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

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16 questions

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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

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

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

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

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

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