Trong không gian O x y z Oxyz O x y z , khoảng cách giữa đường thẳng d d d và mặt phẳng ( α ) \left(\alpha\right) ( α ) là các khẳng định nào sau đây?

d\left(d,\left(\alpha\right)\right)=0 d ( d , ( α ) ) = 0 khi d d nằm trong \left(\alpha\right) ( α ) .

d\left(d,\left(\alpha\right)\right)=d\left(M,\left(\alpha\right)\right),\ \forall M\in d d ( d , ( α ) ) = d ( M , ( α ) ) , ∀ M ∈ d khi d\parallel\left(\alpha\right) d ∥ ( α )

d ( d , ( α ) ) = 0 d\left(d,\left(\alpha\right)\right)=0 d ( d , ( α ) ) = 0 khi d d d cắt ( α ) \left(\alpha\right) ( α ) .

d\left(d,\left(\alpha\right)\right)=d\left(M,d\right),\ \forall M\in\left(\alpha\right) d ( d , ( α ) ) = d ( M , d ) , ∀ M ∈ ( α ) khi d\parallel\left(\alpha\right) d ∥ ( α ) .

Trong không gian O x y z Oxyz O x y z , cho đường thẳng Δ \Delta Δ đi qua M 0 M_0 M 0 và có vectơ chỉ phương u → \overrightarrow{u} u <path d="M0 241v40h399891c-47.3 35.3-84 78-110 128 -16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85 -40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5 -12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67 151.7 139 205zm0 0v40h399900v-40z"> , khoảng cách từ điểm M M đến đường thẳng Δ \Delta Δ là

d\left(M,\Delta\right)=\frac{\left|\left[\overrightarrow{u},\ \overrightarrow{MM_0}\right]\right|}{\left|\overrightarrow{u}\right|} d ( M , Δ ) = ∣ ∣ ∣ u <path d="M0 241v40h399891c-47.3 35.3-84 78-110 128 -16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85 -40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5 -12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67 151.7 139 205zm0 0v40h399900v-40z"> ∣ ∣ ∣ ∣ ∣ ∣ [ u <path d="M0 241v40h399891c-47.3 35.3-84 78-110 128 -16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85 -40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5 -12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67 151.7 139 205zm0 0v40h399900v-40z"> , M M 0 <path d="M0 241v40h399891c-47.3 35.3-84 78-110 128 -16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85 -40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5 -12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67 151.7 139 205zm0 0v40h399900v-40z"> ] ∣ ∣ ∣ .

d ( M , Δ ) = M M 0 d\left(M,\Delta\right)=MM_0 d ( M , Δ ) = M M 0 .

d\left(M,\Delta\right)=\frac{\left[\overrightarrow{u},\ \overrightarrow{MM_0}\right]}{\left|\overrightarrow{u}\right|} d ( M , Δ ) = ∣ ∣ ∣ u <path d="M0 241v40h399891c-47.3 35.3-84 78-110 128 -16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85 -40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5 -12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67 151.7 139 205zm0 0v40h399900v-40z"> ∣ ∣ ∣ [ u <path d="M0 241v40h399891c-47.3 35.3-84 78-110 128 -16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85 -40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5 -12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67 151.7 139 205zm0 0v40h399900v-40z"> , M M 0 <path d="M0 241v40h399891c-47.3 35.3-84 78-110 128 -16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85 -40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5 -12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67 151.7 139 205zm0 0v40h399900v-40z"> ] .

d\left(M,\Delta\right)=\frac{\overrightarrow{u}.\ \overrightarrow{MM_0}}{\left|\overrightarrow{u}\right|} d ( M , Δ ) = ∣ ∣ ∣ u <path d="M0 241v40h399891c-47.3 35.3-84 78-110 128 -16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85 -40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5 -12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67 151.7 139 205zm0 0v40h399900v-40z"> ∣ ∣ ∣ u <path d="M0 241v40h399891c-47.3 35.3-84 78-110 128 -16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85 -40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5 -12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67 151.7 139 205zm0 0v40h399900v-40z"> . M M 0 <path d="M0 241v40h399891c-47.3 35.3-84 78-110 128 -16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85 -40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5 -12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67 151.7 139 205zm0 0v40h399900v-40z"> .

Trong không gian O x y z Oxyz O x y z , công thức tính khoảng cách giữa hai đường thẳng chéo nhau : Δ 1 \Delta_1 Δ 1 đi qua M 1 M_1 M 1 có VTCP u 1 → \overrightarrow{u_1} u 1 <path d="M0 241v40h399891c-47.3 35.3-84 78-110 128 -16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85 -40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5 -12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67 151.7 139 205zm0 0v40h399900v-40z"> , Δ 2 \Delta_2 Δ 2 đi qua M 2 M_2 M 2 có VTCP u 2 → \overrightarrow{u_2} u 2 <path d="M0 241v40h399891c-47.3 35.3-84 78-110 128 -16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85 -40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5 -12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67 151.7 139 205zm0 0v40h399900v-40z"> là

d\left(\Delta_1,\ \Delta_2\right)=\frac{\left|\left[\overrightarrow{u_1},\ \overrightarrow{u_2}\right].\overrightarrow{M_1M_2}\right|}{\left|\left[\overrightarrow{u_1},\ \overrightarrow{u_2}\right]\right|} d ( Δ 1 , Δ 2 ) = ∣ ∣ ∣ [ u 1 <path d="M0 241v40h399891c-47.3 35.3-84 78-110 128 -16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85 -40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5 -12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67 151.7 139 205zm0 0v40h399900v-40z"> , u 2 <path d="M0 241v40h399891c-47.3 35.3-84 78-110 128 -16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85 -40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5 -12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67 151.7 139 205zm0 0v40h399900v-40z"> ] ∣ ∣ ∣ ∣ ∣ ∣ [ u 1 <path d="M0 241v40h399891c-47.3 35.3-84 78-110 128 -16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85 -40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5 -12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67 151.7 139 205zm0 0v40h399900v-40z"> , u 2 <path d="M0 241v40h399891c-47.3 35.3-84 78-110 128 -16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85 -40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5 -12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67 151.7 139 205zm0 0v40h399900v-40z"> ] . M 1 M 2 <path d="M0 241v40h399891c-47.3 35.3-84 78-110 128 -16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85 -40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5 -12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67 151.7 139 205zm0 0v40h399900v-40z"> ∣ ∣ ∣ .

d ( Δ 1 , Δ 2 ) = [ u 1 → , u 2 → ] . M 1 M 2 → [ u 1 → , u 2 → ] d\left(\Delta_1,\ \Delta_2\right)=\frac{\left[\overrightarrow{u_1},\ \overrightarrow{u_2}\right].\overrightarrow{M_1M_2}}{\left[\overrightarrow{u_1},\ \overrightarrow{u_2}\right]} d ( Δ 1 , Δ 2 ) = [ u 1 <path d="M0 241v40h399891c-47.3 35.3-84 78-110 128 -16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85 -40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5 -12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67 151.7 139 205zm0 0v40h399900v-40z"> , u 2 <path d="M0 241v40h399891c-47.3 35.3-84 78-110 128 -16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85 -40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5 -12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67 151.7 139 205zm0 0v40h399900v-40z"> ] [ u 1 <path d="M0 241v40h399891c-47.3 35.3-84 78-110 128 -16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85 -40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5 -12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67 151.7 139 205zm0 0v40h399900v-40z"> , u 2 <path d="M0 241v40h399891c-47.3 35.3-84 78-110 128 -16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85 -40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5 -12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67 151.7 139 205zm0 0v40h399900v-40z"> ] . M 1 M 2 <path d="M0 241v40h399891c-47.3 35.3-84 78-110 128 -16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85 -40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5 -12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67 151.7 139 205zm0 0v40h399900v-40z"> .

d\left(\Delta_1,\ \Delta_2\right)=M_1M_2 d ( Δ 1 , Δ 2 ) = M 1 M 2 .

d\left(\Delta_1,\ \Delta_2\right)=\frac{\left[\overrightarrow{u_1},\ \overrightarrow{u_2}\right].\overrightarrow{M_1M_2}}{\left|\left[\overrightarrow{u_1},\ \overrightarrow{u_2}\right]\right|} d ( Δ 1 , Δ 2 ) = ∣ ∣ ∣ [ u 1 <path d="M0 241v40h399891c-47.3 35.3-84 78-110 128 -16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85 -40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5 -12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67 151.7 139 205zm0 0v40h399900v-40z"> , u 2 <path d="M0 241v40h399891c-47.3 35.3-84 78-110 128 -16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85 -40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5 -12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67 151.7 139 205zm0 0v40h399900v-40z"> ] ∣ ∣ ∣ [ u 1 <path d="M0 241v40h399891c-47.3 35.3-84 78-110 128 -16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85 -40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5 -12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67 151.7 139 205zm0 0v40h399900v-40z"> , u 2 <path d="M0 241v40h399891c-47.3 35.3-84 78-110 128 -16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85 -40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5 -12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67 151.7 139 205zm0 0v40h399900v-40z"> ] . M 1 M 2 <path d="M0 241v40h399891c-47.3 35.3-84 78-110 128 -16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85 -40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5 -12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67 151.7 139 205zm0 0v40h399900v-40z"> .

Trong không gian với hệ tọa độ O x y z Oxyz O x y z , khoảng cách từ điểm M ( 1 ; − 2 ; 3 ) M\left(1;\ -2;\ 3\right) M ( 1 ; − 2 ; 3 ) đến mặt phẳng ( α ) : 3 x + 4 y + 2 z + 4 = 0 \left(\alpha\right):\ 3x+4y+2z+4=0 ( α ) : 3 x + 4 y + 2 z + 4 = 0 là

d\left(M,\left(\alpha\right)\right)=\frac{5}{\sqrt{29}} d ( M , ( α ) ) = 2 9 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z"> 5 .

d ( M , ( α ) ) = 5 3 d\left(M,\left(\alpha\right)\right)=\frac{5}{3} d ( M , ( α ) ) = 3 5 .

d\left(M,\left(\alpha\right)\right)=\frac{5}{29} d ( M , ( α ) ) = 2 9 5 .

d\left(M,\left(\alpha\right)\right)=\frac{1}{\sqrt{29}} d ( M , ( α ) ) = 2 9 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z"> 1 .

Trong không gian với hệ tọa độ O x y z Oxyz O x y z , khoảng cách từ điểm M ( 3 ; 1 ; − 2 ) M\left(3;\ 1;\ -2\right) M ( 3 ; 1 ; − 2 ) đến mặt phẳng ( α ) : 2 x − y + 2 z − 4 = 0 \left(\alpha\right):\ 2x-y+2z-4=0 ( α ) : 2 x − y + 2 z − 4 = 0 là

d\left(M,\left(\alpha\right)\right)=1 d ( M , ( α ) ) = 1 .

d ( M , ( α ) ) = 3 d\left(M,\left(\alpha\right)\right)=3 d ( M , ( α ) ) = 3 .

d\left(M,\left(\alpha\right)\right)=2 d ( M , ( α ) ) = 2 .

d\left(M,\left(\alpha\right)\right)=\frac{1}{3} d ( M , ( α ) ) = 3 1 .

Trong không gian với hệ tọa độ O x y z Oxyz O x y z , khoảng cách giữa hai mặt phẳng ( α ) : x + 2 y + 3 z − 1 = 0 \left(\alpha\right):\ x+2y+3z-1=0 ( α ) : x + 2 y + 3 z − 1 = 0 và ( β ) : x + 2 y + 3 z + 6 = 0 \left(\beta\right):\ x+2y+3z+6=0 ( β ) : x + 2 y + 3 z + 6 = 0 là

d\left(\left(\alpha\right),\left(\beta\right)\right)=\frac{7}{\sqrt{14}} d ( ( α ) , ( β ) ) = 1 4 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z"> 7 .

d ( ( α ) , ( β ) ) = 14 d\left(\left(\alpha\right),\left(\beta\right)\right)=14 d ( ( α ) , ( β ) ) = 1 4 .

d\left(\left(\alpha\right),\left(\beta\right)\right)=\frac{5}{\sqrt{14}} d ( ( α ) , ( β ) ) = 1 4 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z"> 5 .

d\left(\left(\alpha\right),\left(\beta\right)\right)=\frac{-7}{\sqrt{14}} d ( ( α ) , ( β ) ) = 1 4 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z"> − 7 .

Trong không gian với hệ tọa độ O x y z Oxyz O x y z , khoảng cách giữa hai mặt phẳng ( α ) : 2 x − y − 2 z − 9 = 0 \left(\alpha\right):\ 2x-y-2z-9=0 ( α ) : 2 x − y − 2 z − 9 = 0 và ( β ) : 4 x − 2 y − 4 z − 6 = 0 \left(\beta\right):\ 4x-2y-4z-6=0 ( β ) : 4 x − 2 y − 4 z − 6 = 0 là

d\left(\left(\alpha\right),\left(\beta\right)\right)=2 d ( ( α ) , ( β ) ) = 2 .

d ( ( α ) , ( β ) ) = 3 d\left(\left(\alpha\right),\left(\beta\right)\right)=3 d ( ( α ) , ( β ) ) = 3 .

d\left(\left(\alpha\right),\left(\beta\right)\right)=0 d ( ( α ) , ( β ) ) = 0 .

d\left(\left(\alpha\right),\left(\beta\right)\right)=1 d ( ( α ) , ( β ) ) = 1 .

Trong không gian với hệ tọa độ O x y z Oxyz O x y z , khoảng cách giữa đường thẳng Δ : x − 1 2 = y + 2 1 = z − 1 2 \Delta:\frac{x-1}{2}=\frac{y+2}{1}=\frac{z-1}{2} Δ : 2 x − 1 = 1 y + 2 = 2 z − 1 và mặt phẳng ( α ) : 2 x − 2 y − z + 1 = 0 \left(\alpha\right):\ 2x-2y-z+1=0 ( α ) : 2 x − 2 y − z + 1 = 0 là

d\left(\Delta,\left(\alpha\right)\right)=2 d ( Δ , ( α ) ) = 2 .

d ( Δ , ( α ) ) = 2 3 d\left(\Delta,\left(\alpha\right)\right)=\frac{2}{3} d ( Δ , ( α ) ) = 3 2 .

d\left(\Delta,\left(\alpha\right)\right)=\frac{5}{3} d ( Δ , ( α ) ) = 3 5 .

d\left(\Delta,\left(\alpha\right)\right)=\frac{1}{3} d ( Δ , ( α ) ) = 3 1 .

Trong không gian với hệ tọa độ O x y z Oxyz O x y z , khoảng cách giữa đường thẳng Δ : x = 2 + t , y = 5 + 4 t , z = 2 + t ( t ∈ R ) \Delta:\ x=2+t,\ y=5+4t,\ z=2+t\ \left(t\in R\right) Δ : x = 2 + t , y = 5 + 4 t , z = 2 + t ( t ∈ R ) và mặt phẳng ( α ) : 2 x − y + 2 z = 0 \left(\alpha\right):\ 2x-y+2z=0 ( α ) : 2 x − y + 2 z = 0 là

d\left(\Delta,\left(\alpha\right)\right)=1 d ( Δ , ( α ) ) = 1 .

d ( Δ , ( α ) ) = 3 d\left(\Delta,\left(\alpha\right)\right)=3 d ( Δ , ( α ) ) = 3 .

d\left(\Delta,\left(\alpha\right)\right)=2 d ( Δ , ( α ) ) = 2 .

d\left(\Delta,\left(\alpha\right)\right)=0 d ( Δ , ( α ) ) = 0 .

Trong không gian với hệ tọa độ O x y z Oxyz O x y z , khoảng cách từ điểm M ( 2 ; − 1 ; 0 ) M\left(2;-1;\ 0\right) M ( 2 ; − 1 ; 0 ) đường thẳng Δ : x − 3 − 2 = y − 1 = z − 1 1 \Delta:\frac{x-3}{-2}=\frac{y}{-1}=\frac{z-1}{1} Δ : − 2 x − 3 = − 1 y = 1 z − 1 là

d\left(M,\ \Delta\right)=\frac{\sqrt{21}}{3} d ( M , Δ ) = 3 2 1 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z"> .

d ( M , Δ ) = 7 d\left(M,\ \Delta\right)=\sqrt{7} d ( M , Δ ) = 7 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z"> .

d\left(M,\ \Delta\right)=\frac{\sqrt{7}}{3} d ( M , Δ ) = 3 7 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z"> .

d\left(M,\ \Delta\right)=\frac{\sqrt{7}}{2} d ( M , Δ ) = 2 7 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z"> .

Trong không gian với hệ tọa độ O x y z Oxyz O x y z , khoảng cách từ điểm M ( 2 ; − 4 ; − 1 ) M\left(2;-4;\ -1\right) M ( 2 ; − 4 ; − 1 ) đường thẳng Δ : x = t , y = 2 − t , z = 3 + 2 t ( t ∈ R ) \Delta:\ x=t,\ y=2-t,\ z=3+2t\ \left(t\in R\right) Δ : x = t , y = 2 − t , z = 3 + 2 t ( t ∈ R ) là

d\left(M,\ \Delta\right)=2\sqrt{14} d ( M , Δ ) = 2 1 4 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z"> .

d ( M , Δ ) = 6 d\left(M,\ \Delta\right)=\sqrt{6} d ( M , Δ ) = 6 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z"> .

d\left(M,\ \Delta\right)=\sqrt{14} d ( M , Δ ) = 1 4 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z"> .

d\left(M,\ \Delta\right)=2\sqrt{6} d ( M , Δ ) = 2 6 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z"> .

Trong không gian với hệ tọa độ O x y z Oxyz O x y z , khoảng cách giữa hai đường thẳng d : x 3 = y − 3 − 1 = z − 1 1 d:\ \frac{x}{3}=\frac{y-3}{-1}=\frac{z-1}{1} d : 3 x = − 1 y − 3 = 1 z − 1 và Δ : x = 1 + t , y = − 3 − t , z = 2 + 2 t ( t ∈ R ) \Delta:\ x=1+t,\ y=-3-t,\ z=2+2t\ \left(t\in R\right) Δ : x = 1 + t , y = − 3 − t , z = 2 + 2 t ( t ∈ R ) là

\frac{9\sqrt{30}}{10} 1 0 9 3 0 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z"> .

\frac{\sqrt{30}}{3} 3 3 0 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z"> .

\frac{13\sqrt{30}}{30} 3 0 1 3 3 0 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z"> .

Trong không gian với hệ tọa độ O x y z Oxyz O x y z , khoảng cách giữa hai đường thẳng Δ : x − 3 − 2 = y − 1 = z − 1 1 \Delta:\frac{x-3}{-2}=\frac{y}{-1}=\frac{z-1}{1} Δ : − 2 x − 3 = − 1 y = 1 z − 1 và Δ ′ : x − 2 2 = y + 1 1 = z − 1 \Delta':\ \frac{x-2}{2}=\frac{y+1}{1}=\frac{z}{-1} Δ ′ : 2 x − 2 = 1 y + 1 = − 1 z là

21 3 \frac{\sqrt{21}}{3} 3 2 1 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z"> .

7 \sqrt{7} 7 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z"> .

7 3 \frac{\sqrt{7}}{3} 3 7 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z"> .

\frac{\sqrt{7}}{2} 2 7 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z"> .

Oxyz - Khoảng cách

Authored by Nguyễn Phượng

Mathematics

12th Grade

Used 14+ times

AI Actions

Add similar questions

Adjust reading levels

Convert to real-world scenario

Translate activity

More...

Content View

Student View

16 questions

Show all answers

MULTIPLE CHOICE QUESTION

45 sec • 1 pt

Trong không gian $Oxyz$ , khoảng cách từ điểm $M\left(x_0;\ y_0;\ z_0\right)$ đến mặt phẳng $\left(\alpha\right):\ Ax+By+Cz+\ D=0$ là

$d\left(M,\left(\alpha\right)\right)=\frac{\left|Ax_0+By_0+Cz_0+D\right|}{A^2+B^2+C^2}$ .

$d\left(M,\left(\alpha\right)\right)=\frac{\left|Ax_0+By_0+Cz_0\right|}{\sqrt{A^2+B^2+C^2}}$ .

$d\left(M,\left(\alpha\right)\right)=\frac{Ax_0+By_0+Cz_0+D}{\sqrt{A^2+B^2+C^2}}$ .

$d\left(M,\left(\alpha\right)\right)=\frac{\left|Ax_0+By_0+Cz_0+D\right|}{\sqrt{A^2+B^2+C^2}}$ .

MULTIPLE SELECT QUESTION

45 sec • 1 pt

Trong không gian $Oxyz$ , khoảng cách giữa hai mặt phẳng song song $\left(\alpha\right):\ Ax+By+Cz+\ D=0$ và $\left(\beta\right):\ Ax+By+Cz+D'=0$ là các khẳng định nào sau đây?

$d\left(\left(\alpha\right),\left(\beta\right)\right)=\frac{D-D'}{\sqrt{A^2+B^2+C^{22}}}$ với $M\in\left(\beta\right)$ .

$d\left(\left(\alpha\right),\left(\beta\right)\right)=d\left(M,\ \left(\beta\right)\right)$ với $M\in\left(\alpha\right)$ .

$d\left(\left(\alpha\right),\left(\beta\right)\right)=\frac{\left|D-D'\right|}{\sqrt{A^2+B^2+C^2}}$ .

$d\left(\left(\alpha\right),\left(\beta\right)\right)=d\left(M,\ \left(\beta\right)\right)$ với $M\in\left(\alpha\right)$ .

MULTIPLE SELECT QUESTION

45 sec • 1 pt

Trong không gian $Oxyz$ , khoảng cách giữa đường thẳng $d$ và mặt phẳng $\left(\alpha\right)$ là các khẳng định nào sau đây?

$d\left(d,\left(\alpha\right)\right)=0$ khi $d$ cắt $\left(\alpha\right)$ .

$d\left(d,\left(\alpha\right)\right)=d\left(M,\left(\alpha\right)\right),\ \forall M\in d$ khi $d\parallel\left(\alpha\right)$

$d\left(d,\left(\alpha\right)\right)=d\left(M,d\right),\ \forall M\in\left(\alpha\right)$ khi $d\parallel\left(\alpha\right)$ .

$d\left(d,\left(\alpha\right)\right)=0$ khi $d$ nằm trong $\left(\alpha\right)$ .

MULTIPLE CHOICE QUESTION

45 sec • 1 pt

Trong không gian $Oxyz$ , cho đường thẳng $\Delta$ đi qua $M_0$ và có vectơ chỉ phương $\overrightarrow{u}$ , khoảng cách từ điểm $M$ đến đường thẳng $\Delta$ là

$d\left(M,\Delta\right)=MM_0$ .

$d\left(M,\Delta\right)=\frac{\left|\left[\overrightarrow{u},\ \overrightarrow{MM_0}\right]\right|}{\left|\overrightarrow{u}\right|}$ .

$d\left(M,\Delta\right)=\frac{\left[\overrightarrow{u},\ \overrightarrow{MM_0}\right]}{\left|\overrightarrow{u}\right|}$ .

$d\left(M,\Delta\right)=\frac{\overrightarrow{u}.\ \overrightarrow{MM_0}}{\left|\overrightarrow{u}\right|}$ .

MULTIPLE SELECT QUESTION

45 sec • 1 pt

Trong không gian $Oxyz$ , các công thức tính khoảng cách giữa hai đường thẳng song song $\Delta_1,\ \Delta_2$ là

$d\left(\Delta_1,\ \Delta_2\right)=M_1M_2,\ \forall M_1\in\Delta_1,\ \ \forall M_2\in\Delta_2$ .

$d\left(\Delta_1,\ \Delta_2\right)=d\left(M,\ \Delta_2\right),\ \forall M\in\Delta_1$ khi $\Delta_1\parallel\Delta_2$ .

$d\left(\Delta_1,\ \Delta_2\right)=d\left(M,\ \Delta_1\right),\ \forall M\in\Delta_2$ khi $\Delta_1\parallel\Delta_2$ .

$d\left(\Delta_1,\ \Delta_2\right)=0$ khi $\Delta_1\equiv\Delta_2$ hoặc $\Delta_1$ cắt $\Delta_2$ .

MULTIPLE CHOICE QUESTION

45 sec • 1 pt

Trong không gian $Oxyz$ , công thức tính khoảng cách giữa hai đường thẳng chéo nhau : $\Delta_1$ đi qua $M_1$ có VTCP $\overrightarrow{u_1}$ , $\Delta_2$ đi qua $M_2$ có VTCP $\overrightarrow{u_2}$ là

$d\left(\Delta_1,\ \Delta_2\right)=\frac{\left[\overrightarrow{u_1},\ \overrightarrow{u_2}\right].\overrightarrow{M_1M_2}}{\left[\overrightarrow{u_1},\ \overrightarrow{u_2}\right]}$ .

$d\left(\Delta_1,\ \Delta_2\right)=M_1M_2$ .

$d\left(\Delta_1,\ \Delta_2\right)=\frac{\left|\left[\overrightarrow{u_1},\ \overrightarrow{u_2}\right].\overrightarrow{M_1M_2}\right|}{\left|\left[\overrightarrow{u_1},\ \overrightarrow{u_2}\right]\right|}$ .

$d\left(\Delta_1,\ \Delta_2\right)=\frac{\left[\overrightarrow{u_1},\ \overrightarrow{u_2}\right].\overrightarrow{M_1M_2}}{\left|\left[\overrightarrow{u_1},\ \overrightarrow{u_2}\right]\right|}$ .

MULTIPLE CHOICE QUESTION

45 sec • 1 pt

Trong không gian với hệ tọa độ $Oxyz$ , khoảng cách từ điểm $M\left(1;\ -2;\ 3\right)$ đến mặt phẳng $\left(\alpha\right):\ 3x+4y+2z+4=0$ là

$d\left(M,\left(\alpha\right)\right)=\frac{5}{3}$ .

$d\left(M,\left(\alpha\right)\right)=\frac{5}{29}$ .

$d\left(M,\left(\alpha\right)\right)=\frac{5}{\sqrt{29}}$ .

$d\left(M,\left(\alpha\right)\right)=\frac{1}{\sqrt{29}}$ .

MULTIPLE CHOICE QUESTION

45 sec • 1 pt

Trong không gian với hệ tọa độ $Oxyz$ , khoảng cách từ điểm $M\left(3;\ 1;\ -2\right)$ đến mặt phẳng $\left(\alpha\right):\ 2x-y+2z-4=0$ là

$d\left(M,\left(\alpha\right)\right)=3$ .

$d\left(M,\left(\alpha\right)\right)=2$ .

$d\left(M,\left(\alpha\right)\right)=\frac{1}{3}$ .

$d\left(M,\left(\alpha\right)\right)=1$ .

MULTIPLE CHOICE QUESTION

45 sec • 1 pt

Trong không gian với hệ tọa độ $Oxyz$ , khoảng cách giữa hai mặt phẳng $\left(\alpha\right):\ x+2y+3z-1=0$ và $\left(\beta\right):\ x+2y+3z+6=0$ là

$d\left(\left(\alpha\right),\left(\beta\right)\right)=14$ .

$d\left(\left(\alpha\right),\left(\beta\right)\right)=\frac{5}{\sqrt{14}}$ .

$d\left(\left(\alpha\right),\left(\beta\right)\right)=\frac{7}{\sqrt{14}}$ .

$d\left(\left(\alpha\right),\left(\beta\right)\right)=\frac{-7}{\sqrt{14}}$ .

10.

MULTIPLE CHOICE QUESTION

45 sec • 1 pt

Trong không gian với hệ tọa độ $Oxyz$ , khoảng cách giữa hai mặt phẳng $\left(\alpha\right):\ 2x-y-2z-9=0$ và $\left(\beta\right):\ 4x-2y-4z-6=0$ là

$d\left(\left(\alpha\right),\left(\beta\right)\right)=3$ .

$d\left(\left(\alpha\right),\left(\beta\right)\right)=0$ .

$d\left(\left(\alpha\right),\left(\beta\right)\right)=1$ .

$d\left(\left(\alpha\right),\left(\beta\right)\right)=2$ .

Access all questions and much more by creating a free account

Create resources

Host any resource

Get auto-graded reports

Continue with Google

Continue with Email

Continue with Classlink

Continue with Clever

or continue with

Microsoft

Apple

Others

Already have an account?

Similar Resources on Wayground

11 questions

Կրճատ բազմապատկման բանաձևեր ( Ձևավորող)

Quiz

•

7th - 12th Grade

12 questions

REVISION: TOPIC 3 & TOPIC 4

Quiz

•

12th Grade

17 questions

Unit 1

Quiz

•

9th Grade - University

15 questions

Test1

Quiz

•

10th - 12th Grade

20 questions

Matematika Peminatan - XI

Quiz

•

10th - 12th Grade

11 questions

PH FUNGSI INVERS

Quiz

•

12th Grade

15 questions

TERCERO BGU UET

Quiz

•

12th Grade

15 questions

Studio di funzione

Quiz

•

12th Grade - University

Popular Resources on Wayground

15 questions

Fractions on a Number Line

Quiz

•

3rd Grade

10 questions

Probability Practice

Quiz

•

4th Grade

15 questions

Probability on Number LIne

Quiz

•

4th Grade

20 questions

Equivalent Fractions

Quiz

•

3rd Grade

25 questions

Multiplication Facts

Quiz

•

5th Grade

$fractions$

22 questions

fractions

Quiz

•

3rd Grade

6 questions

Appropriate Chromebook Usage

Lesson

•

7th Grade

10 questions

Greek Bases tele and phon

Quiz

•

6th - 8th Grade

Discover more resources for Mathematics

23 questions

TSI Math Vocabulary

Quiz

•

10th - 12th Grade

20 questions

SSS/SAS

Quiz

•

9th - 12th Grade

15 questions

Exponential Growth and Decay Word Problems Practice

Quiz

•

9th - 12th Grade

20 questions

9.1 & 9.2 Exponential Growth and Decay

Quiz

•

12th Grade

13 questions

Identify Transformations in Geometry

Quiz

•

8th - 12th Grade

20 questions

Quadratic Transformations Review

Quiz

•

9th - 12th Grade

10 questions

Naming Angles Formed By Two Lines And A Transversal

Quiz

•

8th - 12th Grade

10 questions

Intro to Rational Graphs

Quiz

•

9th - 12th Grade

Oxyz - Khoảng cách

Trong không gian OxyzOxyzOxyz , khoảng cách từ điểm M(x0; y0; z0)M\left(x_0;\ y_0;\ z_0\right)M(x0​; y0​; z0​) đến mặt phẳng (α): Ax+By+Cz+ D=0\left(\alpha\right):\ Ax+By+Cz+\ D=0(α): Ax+By+Cz+ D=0 là

Trong không gian OxyzOxyzOxyz , khoảng cách giữa hai mặt phẳng song song (α): Ax+By+Cz+ D=0\left(\alpha\right):\ Ax+By+Cz+\ D=0(α): Ax+By+Cz+ D=0 và (β): Ax+By+Cz+D′=0\left(\beta\right):\ Ax+By+Cz+D'=0(β): Ax+By+Cz+D′=0 là các khẳng định nào sau đây?

Trong không gian OxyzOxyzOxyz , khoảng cách giữa đường thẳng ddd và mặt phẳng (α)\left(\alpha\right)(α) là các khẳng định nào sau đây?

Trong không gian OxyzOxyzOxyz , cho đường thẳng Δ\DeltaΔ đi qua M0M_0M0​ và có vectơ chỉ phương u→\overrightarrow{u}u , khoảng cách từ điểm MM đến đường thẳng Δ\DeltaΔ là

Trong không gian OxyzOxyzOxyz , các công thức tính khoảng cách giữa hai đường thẳng song song Δ1, Δ2\Delta_1,\ \Delta_2Δ1​, Δ2​ là

Trong không gian OxyzOxyzOxyz , công thức tính khoảng cách giữa hai đường thẳng chéo nhau : Δ1\Delta_1Δ1​ đi qua M1M_1M1​ có VTCP u1→\overrightarrow{u_1}u1​​ , Δ2\Delta_2Δ2​ đi qua M2M_2M2​ có VTCP u2→\overrightarrow{u_2}u2​​ là

Trong không gian với hệ tọa độ OxyzOxyzOxyz , khoảng cách từ điểm M(1; −2; 3)M\left(1;\ -2;\ 3\right)M(1; −2; 3) đến mặt phẳng (α): 3x+4y+2z+4=0\left(\alpha\right):\ 3x+4y+2z+4=0(α): 3x+4y+2z+4=0 là

Trong không gian với hệ tọa độ OxyzOxyzOxyz , khoảng cách từ điểm M(3; 1; −2)M\left(3;\ 1;\ -2\right)M(3; 1; −2) đến mặt phẳng (α): 2x−y+2z−4=0\left(\alpha\right):\ 2x-y+2z-4=0(α): 2x−y+2z−4=0 là

Trong không gian với hệ tọa độ OxyzOxyzOxyz , khoảng cách giữa hai mặt phẳng (α): x+2y+3z−1=0\left(\alpha\right):\ x+2y+3z-1=0(α): x+2y+3z−1=0 và (β): x+2y+3z+6=0\left(\beta\right):\ x+2y+3z+6=0(β): x+2y+3z+6=0 là

Trong không gian với hệ tọa độ OxyzOxyzOxyz , khoảng cách giữa hai mặt phẳng (α): 2x−y−2z−9=0\left(\alpha\right):\ 2x-y-2z-9=0(α): 2x−y−2z−9=0 và (β): 4x−2y−4z−6=0\left(\beta\right):\ 4x-2y-4z-6=0(β): 4x−2y−4z−6=0 là

Access all questions and much more by creating a free account

Similar Resources on Wayground

Popular Resources on Wayground

Discover more resources for Mathematics

Trong không gian $Oxyz$ , khoảng cách từ điểm $M\left(x_0;\ y_0;\ z_0\right)$ đến mặt phẳng $\left(\alpha\right):\ Ax+By+Cz+\ D=0$ là

Trong không gian $Oxyz$ , khoảng cách giữa hai mặt phẳng song song $\left(\alpha\right):\ Ax+By+Cz+\ D=0$ và $\left(\beta\right):\ Ax+By+Cz+D'=0$ là các khẳng định nào sau đây?

Trong không gian $Oxyz$ , khoảng cách giữa đường thẳng $d$ và mặt phẳng $\left(\alpha\right)$ là các khẳng định nào sau đây?

Trong không gian $Oxyz$ , cho đường thẳng $\Delta$ đi qua $M_0$ và có vectơ chỉ phương $\overrightarrow{u}$ , khoảng cách từ điểm $M$ đến đường thẳng $\Delta$ là

Trong không gian $Oxyz$ , các công thức tính khoảng cách giữa hai đường thẳng song song $\Delta_1,\ \Delta_2$ là

Trong không gian $Oxyz$ , công thức tính khoảng cách giữa hai đường thẳng chéo nhau : $\Delta_1$ đi qua $M_1$ có VTCP $\overrightarrow{u_1}$ , $\Delta_2$ đi qua $M_2$ có VTCP $\overrightarrow{u_2}$ là

Trong không gian với hệ tọa độ $Oxyz$ , khoảng cách từ điểm $M\left(1;\ -2;\ 3\right)$ đến mặt phẳng $\left(\alpha\right):\ 3x+4y+2z+4=0$ là

Trong không gian với hệ tọa độ $Oxyz$ , khoảng cách từ điểm $M\left(3;\ 1;\ -2\right)$ đến mặt phẳng $\left(\alpha\right):\ 2x-y+2z-4=0$ là

Trong không gian với hệ tọa độ $Oxyz$ , khoảng cách giữa hai mặt phẳng $\left(\alpha\right):\ x+2y+3z-1=0$ và $\left(\beta\right):\ x+2y+3z+6=0$ là

Trong không gian với hệ tọa độ $Oxyz$ , khoảng cách giữa hai mặt phẳng $\left(\alpha\right):\ 2x-y-2z-9=0$ và $\left(\beta\right):\ 4x-2y-4z-6=0$ là