Find the line through the point [ 1 , 3 , 3 ] and parallel to the vector [ 1 , 4 , 2 ].
[ x , y , z ] = [ 1 + t , 3 + 5t , 3 + 3t ] [ x , y , z ] = [ 1 + t , 3 + 4t , 3 + 2t ] [ x , y , z ] = [ 2 + t , 3 + 4t , 4 + 2t ] [ x , y , z ] = [ 2 + t , 3 + 5t , 4 + 3t ]
Find the line through the points [ − 1 , 0 , 0 ] and [ 2 , 1 , − 1 ] .
[ x , y , z ] = [ − 1 + 3t , 2t , − t ] [ x , y , z ] = [ 3t , t , 1 − t ] [ x , y , z ] = [ 3t , 2t , 1 − t ] [ x , y , z ] = [ − 1 + 3t , t , − t ]
Find the line through the point [ − 3 , − 2 , − 3 ] and perpendicular to the plane 1 + x + 2z = 0.
[ x , y , z ] = [ − 3 + t , − 2 , − 3 + 2t ] [ x , y , z ] = [ − 3 + t , − 2 + t , − 3 + 3t ] [ x , y , z ] = [ − 2 + t , − 2 + t , − 3 + 3t ] [ x , y , z ] = [ − 2 + t , − 2 , − 3 + 2t ]
Find the line of intersection of the planes −x + 3y = 0 and −16 − 3x − y + 2z = 0 using the point [ − 3 , − 1 , 3 ] of intersection.
[ x , y , z ] = [ − 3 − t , − 1 + 3t , 3 ]
[ x , y , z ] = [ − 3 − 3t , − 1 − t , 3 + 2t ]
Find the plane through the point [ 3 , 2 , − 4 ] and perpendicular to the vector [ 4 , − 1 , − 1 ].
−18 + 4x − y − z = 0 −20 + 4x − z = 0 −16 + 4x − z = 0 −14 + 4x − y − z = 0
Find the plane through the points [ − 2 , 0 , − 3 ], [ − 2 , 1 , − 4 ], and [ − 4 , − 1 , − 2 ].
4 + 3y + 2z = 0 6 + 3y + 2z = 0 4 + 2y + 2z = 0 6 + 2y + 2z = 0
Find the plane through the point [ − 2 , 1 , − 3 ] and containing the line [ x , y , z ] = [ 1 + t , 1 − 4t , − 2 + 2t ].
23 − 4x + 5y + 12z = 0 27 − 4x + 5y + 12z = 0 26 − 4x + 6y + 12z = 0 22 − 4x + 6y + 12z = 0
Find the distance between the point [ 1 , 3 , − 1 ] and the planes −3 − 3x + 2y + 4z = 0.
0
Find the distance between the planes 16 − 3x − 2y − 2z = 0 and −13 − 3x − 2y − 2z = 0.
Find the tangent line to
at
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[ x , y , z ] = [ 1 + 2t , 1 + t , 1 + 3t ] [ x , y , z ] = [ 1 + 2t , 1 + 2t , 1 + 3t ] [ x , y , z ] = [ 2 + 2t , 1 + 2t , 1 + 3t ] [ x , y , z ] = [ 2 + 2t , 1 + t , 1 + 3t ]
Let
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Suppose that
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[ − 1 , − 1 , 4 ] [ − 2 , − 1 , 3 ] [ − 2 , 1 , 5 ] [ − 3 , 1 , 4 ]
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−6 −5 −5 −8