Find the distance between the planes −2 − x + 2y − 3z = 0 and 9 − x + 2y − 3z = 0.
Find the plane through the points [ 3 , 0 , 1 ], [ 5 , − 2 , 3 ], and [ 5 , 0 , 2 ].
1 − 2x + 3y + 5z = 0 3 − 2x + 3y + 5z = 0 4 − 2x + 2y + 4z = 0 2 − 2x + 2y + 4z = 0
Find the line of intersection of the planes −3 + 3x − y − 2z = 0 and −1 + 3x + y − 2z = 0 using the point [ 0 , − 1 , − 1 ] of intersection.
[ x , y , z ] = [ 4t , − 1 + t , − 1 + 7t ] [ x , y , z ] = [ 4t , − 1 , − 1 + 6t ] [ x , y , z ] = [ 3t , − 1 + t , − 1 − 2t ] [ x , y , z ] = [ 3t , − 1 − t , − 1 − 2t ]
Find the line through the point [ − 2 , − 1 , 3 ] and parallel to the vector [ − 2 , 2 , 0 ].
[ x , y , z ] = [ − 2 − 2t , − 1 + 3t , 3 ] [ x , y , z ] = [ − 2 − 2t , − 1 + 2t , 3 ] [ x , y , z ] = [ − 1 − 2t , − 1 + 2t , 4 ] [ x , y , z ] = [ − 1 − 2t , − 1 + 3t , 4 ]
Find the tangent line to
at
.
[ x , y , z ] = [ 1 − 2t , − 1 + 3t , − 1 + t ] [ x , y , z ] = [ 2 − 2t , − 1 + 3t , t ] [ x , y , z ] = [ 2 − 2t , − 1 + 4t , t ] [ x , y , z ] = [ 1 − 2t , − 1 + 4t , − 1 + t ]
Find the plane through the point [ 0 , − 3 , 1 ]
and containing the line
.
22 − 4x + 10y + 12z = 0 18 − 4x + 10y + 12z = 0 21 − 4x + 11y + 12z = 0 25 − 4x + 11y + 12z = 0
Let
.
Suppose that
,
,
, and
.
Then find
.
[ − 5 , 0 , 0 ] [ − 3 , 0 , − 2 ] [ − 6 , 3 , 1 ] [ − 4 , 3 , − 1 ]
Find the plane through the point [ − 4 , − 3 , − 4 ] and perpendicular to the vector [ 3 , 4 , − 3 ].
9 + 3x + 4y − 3z = 0 15 + 3x + 5y − 3z = 0 12 + 3x + 4y − 3z = 0 12 + 3x + 5y − 3z = 0
Let
.
Suppose that
,
,
, and
.
Then find
.
2 −1 0 1
Find the distance between the point [ 4 , − 3 , − 4 ] and the planes 7 + 4x − 3y − 3z = 0.