Trong không gian Oxyz, cho mặt phẳng ( P ) : 2 x − 3 y + 4 z − 2020 = 0 \left(P\right):2x-3y+4z-2020=0 ( P ) : 2 x − 3 y + 4 z − 2 0 2 0 = 0 . Véctơ nào sau đây là một véctơ pháp tuyến của mặt phẳng (P) ?

n → = ( 2 ; − 3 ; 4 ) \overrightarrow{n}=\left(2;-3;4\right) n <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"> = ( 2 ; − 3 ; 4 )

n → = ( − 2 ; − 3 ; 4 ) \overrightarrow{n}=\left(-2;-3;4\right) n <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"> = ( − 2 ; − 3 ; 4 )

n → = ( 2 ; 3 ; − 4 ) \overrightarrow{n}=\left(2;3;-4\right) n <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"> = ( 2 ; 3 ; − 4 )

n → = ( − 2 ; 3 ; 4 ) \overrightarrow{n}=\left(-2;3;4\right) n <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"> = ( − 2 ; 3 ; 4 )

Vậy mặt phẳng (Q) đi qua điểm M ( 2 ; − 1 ; 3 ) M\left(2;-1;\ 3\right) M ( 2 ; − 1 ; 3 ) và có VTPT n → = ( 3 ; 2 ; − 1 ) \overrightarrow{n\ }=\left(3;\ 2;-1\right) n <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"> = ( 3 ; 2 ; − 1 ) có phương trình là:

3 x + 2 y − z − 1 = 0 3x+2y-z-1=0 3 x + 2 y − z − 1 = 0

2 x − y + 3 z − 1 = 0 2x-y+3z-1=0 2 x − y + 3 z − 1 = 0

3 x + 2 y − z + 1 = 0 3x+2y-z+1=0 3 x + 2 y − z + 1 = 0

2 x − y + 3 z + 1 = 0 2x-y+3z+1=0 2 x − y + 3 z + 1 = 0

Mặt phẳng x − 5 y + 6 = 0 x-5y+6=0 x − 5 y + 6 = 0 có một VTPT là:

n → = ( 1 ; − 5 ; 0 ) \overrightarrow{n\ }=\left(1;-5;\ 0\right) n <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"> = ( 1 ; − 5 ; 0 )

n → = ( 1 ; − 5 ; 6 ) \overrightarrow{n\ }=\left(1;-5;\ 6\right) n <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"> = ( 1 ; − 5 ; 6 )

n → = ( 1 ; 5 ; 0 ) \overrightarrow{n\ }=\left(1;\ 5;\ 0\right) n <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"> = ( 1 ; 5 ; 0 )

n → = ( 1 ; 5 ; 6 ) \overrightarrow{n\ }=\left(1;\ 5;\ 6\right) n <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"> = ( 1 ; 5 ; 6 )

Mặt phẳng đi qua 3 điểm A ( 1 ; 0 ; − 2 ) , B ( 2 ; 3 ; − 1 ) , C ( 0 ; 1 ; 1 ) A\left(1;0;-2\right),\ B\left(2;3;-1\right),\ C\left(0;1;1\right) A ( 1 ; 0 ; − 2 ) , B ( 2 ; 3 ; − 1 ) , C ( 0 ; 1 ; 1 ) có một VTPT là:

n → = ( 2 ; − 1 ; 1 ) \overrightarrow{n\ }=\left(2;-1;\ 1\right) n <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"> = ( 2 ; − 1 ; 1 )

n → = ( 8 ; 4 ; − 4 ) \overrightarrow{n\ }=\left(8;\ 4;-4\right) n <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"> = ( 8 ; 4 ; − 4 )

n → = ( 0 ; 4 ; 4 ) \overrightarrow{n\ }=\left(0;\ 4;\ 4\right) n <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"> = ( 0 ; 4 ; 4 )

n → = ( 1 ; 1 ; − 1 ) \overrightarrow{n\ }=\left(1;1;-1\right) n <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"> = ( 1 ; 1 ; − 1 )

Mặt phẳng nào sau đây đi qua gốc tọa độ?

2 x − 5 y + z = 0 2x-5y+z=0 2 x − 5 y + z = 0

3 x − 7 = 0 3x-7=0 3 x − 7 = 0

2 x + y − 1 = 0 2x+y-1=0 2 x + y − 1 = 0

x + y + z − 3 = 0 x+y+z-3=0 x + y + z − 3 = 0

Mặt phẳng ( α ) : 2 x − 3 y + 5 z − 20 = 0 \left(\alpha\right):\ \ 2x-3y+5z-20=0 ( α ) : 2 x − 3 y + 5 z − 2 0 = 0 có một vectơ pháp tuyến là:

n → = ( − 4 ; 6 ; − 10 ) \overrightarrow{n\ }=\left(-4;6;-10\right) n <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"> = ( − 4 ; 6 ; − 1 0 )

n → = ( 4 ; 6 ; − 10 ) \overrightarrow{n\ }=\left(4;6;-10\right) n <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"> = ( 4 ; 6 ; − 1 0 )

n → = ( 2 ; 3 ; 5 ) \overrightarrow{n\ }=\left(2;3;5\right) n <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"> = ( 2 ; 3 ; 5 )

n → = ( − 2 ; − 3 ; − 5 ) \overrightarrow{n\ }=\left(-2;-3;-5\right) n <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"> = ( − 2 ; − 3 ; − 5 )

Mặt phẳng ( α ) \left(\alpha\right) ( α ) đi qua A ( 2 ; − 1 ; 5 ) A\left(2;-1;5\right) A ( 2 ; − 1 ; 5 ) và chứa trục Ox có vectơ pháp tuyến là n → = ( A ; B ; C ) \overrightarrow{n\ }=\left(A;B;C\right) n <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"> = ( A ; B ; C ) . Khi đó tỉ số B C = ? \frac{B}{C}=? C B = ?

B C = 1 5 \frac{B}{C}=\frac{1}{5} C B = 5 1

B C = − 5 \frac{B}{C}=-5 C B = − 5

B C = − 1 5 \frac{B}{C}=-\frac{1}{5} C B = − 5 1

Mặt phẳng đi qua ba điểm A ( 2 ; 0 ; 0 ) , B ( 0 ; 0 ; − 5 ) , C ( 0 ; 3 ; 0 ) A\left(2;0;0\right),\ B\left(0;0;-5\right),\ C\left(0;3;0\right) A ( 2 ; 0 ; 0 ) , B ( 0 ; 0 ; − 5 ) , C ( 0 ; 3 ; 0 ) có phương trình là:

x 2 + y 3 + z − 5 = 1 \frac{x}{2}+\frac{y}{3}+\frac{z}{-5}=1 2 x + 3 y + − 5 z = 1

x 2 + y − 5 + z 3 = 1 \frac{x}{2}+\frac{y}{-5}+\frac{z}{3}=1 2 x + − 5 y + 3 z = 1

x 2 + y 3 + z − 5 = 0 \frac{x}{2}+\frac{y}{3}+\frac{z}{-5}=0 2 x + 3 y + − 5 z = 0

x 2 + y − 5 + z 3 = 0 \frac{x}{2}+\frac{y}{-5}+\frac{z}{3}=0 2 x + − 5 y + 3 z = 0

Điểm nào sau đây thuộc mặt phẳng x − 3 y + 2 z + 1 = 0 x-3y+2z+1=0 x − 3 y + 2 z + 1 = 0 ?

N ( 0 ; 1 ; 1 ) N\left(0;1;1\right) N ( 0 ; 1 ; 1 )

Q ( 2 ; 0 ; − 1 ) Q\left(2;0;-1\right) Q ( 2 ; 0 ; − 1 )

M ( 3 ; 1 ; 0 ) M\left(3;1;0\right) M ( 3 ; 1 ; 0 )

P ( 1 ; 1 ; 1 ) P\left(1;1;1\right) P ( 1 ; 1 ; 1 )

Mặt phẳng nào sau đây song song với trục Oz ?

2 x + y − 1 = 0 2x+y-1=0 2 x + y − 1 = 0

2 x − 5 y + z = 0 2x-5y+z=0 2 x − 5 y + z = 0

3 z − 7 = 0 3z-7=0 3 z − 7 = 0

Xét vị trí tương đối của hai mặt phẳng ( α ) : 3 x − 2 y + z − 1 = 0 \left(\alpha\right):\ 3x-2y+z-1=0 ( α ) : 3 x − 2 y + z − 1 = 0 và ( β ) : − 6 x + 4 y − 2 z − 2 = 0 \left(\beta\right):-6x+4y-2z-2=0 ( β ) : − 6 x + 4 y − 2 z − 2 = 0

Cắt nhau, nhưng không vuông góc với nhau

Cắt nhau và vuông góc với nhau

Mặt phẳng ( P ) đi qua điểm M ( 2 ; − 1 ; 5 ) M\left(2;-1;5\right) M ( 2 ; − 1 ; 5 ) và song song với mặt phẳng ( Q ) : 2 x − 2 y + z − 3 = 0 \left(Q\right):\ 2x-2y+z-3=0 ( Q ) : 2 x − 2 y + z − 3 = 0 có phương trình là

( P ) : 2 x − 2 y + z − 11 = 0 \left(P\right):\ 2x-2y+z-11=0 ( P ) : 2 x − 2 y + z − 1 1 = 0

( P ) : 2 x − 2 y + z + 11 = 0 \left(P\right):\ 2x-2y+z+11=0 ( P ) : 2 x − 2 y + z + 1 1 = 0

( P ) : 2 x − 2 y + z − 8 = 0 \left(P\right):\ 2x-2y+z-8=0 ( P ) : 2 x − 2 y + z − 8 = 0

( P ) : 2 x − 2 y + z + 8 = 0 \left(P\right):\ 2x-2y+z+8=0 ( P ) : 2 x − 2 y + z + 8 = 0

Cho hai điểm A ( 1 ; 1 ; 1 ) , B ( 1 ; 3 ; − 5 ) A\left(1;1;1\right),\ B\left(1;3;-5\right) A ( 1 ; 1 ; 1 ) , B ( 1 ; 3 ; − 5 ) . Mặt phẳng trung trực của đoạn thẳng AB có phương trình là:

y − 3 z − 8 = 0 y-3z-8=0 y − 3 z − 8 = 0

y − 2 z + 2 = 0 y-2z+2=0 y − 2 z + 2 = 0

y − 3 z + 4 = 0 y-3z+4=0 y − 3 z + 4 = 0

y − 2 z − 6 = 0 y-2z-6=0 y − 2 z − 6 = 0

PHƯƠG TRÌNH MẶT PHẲNG

Authored by Tâm Nguyễn

Mathematics

12th Grade

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Mặt phẳng đi qua 3 điểm $A\left(1;0;-2\right),\ B\left(2;3;-1\right),\ C\left(0;1;1\right)$ có một VTPT là:

$\overrightarrow{n\ }=\left(2;-1;\ 1\right)$

$\overrightarrow{n\ }=\left(8;\ 4;-4\right)$

$\overrightarrow{n\ }=\left(0;\ 4;\ 4\right)$

$\overrightarrow{n\ }=\left(1;1;-1\right)$

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PHƯƠG TRÌNH MẶT PHẲNG

Trong không gian Oxyz, cho mặt phẳng (P):2x−3y+4z−2020=0\left(P\right):2x-3y+4z-2020=0(P):2x−3y+4z−2020=0 . Véctơ nào sau đây là một véctơ pháp tuyến của mặt phẳng (P) ?

Mỗi mặt phẳng có bao nhiêu vectơ pháp tuyến?

Các VTPT của cùng một mặt phẳng có mối liên hệ gì với nhau?

Tích có hướng của hai vec tơ a →\overrightarrow{a\ }a và b →\overrightarrow{b\ }b là một vectơ:

Nếu có hai vectơ a →\overrightarrow{a\ }a và b →\overrightarrow{b\ }b không cùng phương, có giá song song hoặc nằm trong mặt phẳng (P) thì tích có hướng của hai vectơ đó:

Tồn tại bao nhiêu mặt phẳng đi qua điểm M(x0; y0; z0)M\left(x_0;\ y_0;\ z_0\right)M(x0​; y0​; z0​) và nhận vectơ n →=(A; B; C)\overrightarrow{n\ }=\left(A;\ B;\ C\right)n =(A; B; C) làm VTPT?

Vậy mặt phẳng (Q) đi qua điểm M(2;−1; 3)M\left(2;-1;\ 3\right)M(2;−1; 3) và có VTPT n →=(3; 2;−1)\overrightarrow{n\ }=\left(3;\ 2;-1\right)n =(3; 2;−1) có phương trình là:

Mặt phẳng x−5y+6=0x-5y+6=0x−5y+6=0 có một VTPT là:

Mặt phẳng đi qua 3 điểm không thẳng hàng A, B, C có VTPT là vectơ nào sau đây?

Mặt phẳng đi qua 3 điểm A(1;0;−2), B(2;3;−1), C(0;1;1)A\left(1;0;-2\right),\ B\left(2;3;-1\right),\ C\left(0;1;1\right)A(1;0;−2), B(2;3;−1), C(0;1;1) có một VTPT là:

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Trong không gian Oxyz, cho mặt phẳng $\left(P\right):2x-3y+4z-2020=0$ . Véctơ nào sau đây là một véctơ pháp tuyến của mặt phẳng (P) ?

Tích có hướng của hai vec tơ $\overrightarrow{a\ }$ và $\overrightarrow{b\ }$ là một vectơ:

Nếu có hai vectơ $\overrightarrow{a\ }$ và $\overrightarrow{b\ }$ không cùng phương, có giá song song hoặc nằm trong mặt phẳng (P) thì tích có hướng của hai vectơ đó:

Tồn tại bao nhiêu mặt phẳng đi qua điểm $M\left(x_0;\ y_0;\ z_0\right)$ và nhận vectơ $\overrightarrow{n\ }=\left(A;\ B;\ C\right)$ làm VTPT?

Vậy mặt phẳng (Q) đi qua điểm $M\left(2;-1;\ 3\right)$ và có VTPT $\overrightarrow{n\ }=\left(3;\ 2;-1\right)$ có phương trình là:

Mặt phẳng $x-5y+6=0$ có một VTPT là:

Mặt phẳng đi qua 3 điểm $A\left(1;0;-2\right),\ B\left(2;3;-1\right),\ C\left(0;1;1\right)$ có một VTPT là: