Vectors, parallel and perpendicular 10 points?

Question

Find a vector which is perpendicular to the vector (-3,2,5) and parallel to the plane of the vectors 4i+2j-k and 5i-j+k.

Please show how you do this because I am eager to learn.

Answer 1

Find a vector w, which is perpendicular to the vector u = (-3,2,5) and parallel to the plane of the vectors v1 = 4i+2j-k and v2 = 5i-j+k

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If the desired vector is parallel to a plane, then it is orthogonal to the normal vector n, of the plane.  Take the cross product.

n = <4, 2, -1> X <5, -1, 1> = <1, -9, -14>

The desired vector w is orthogonal to both u and n.  Take the cross product.

w = u X n = <-3, 2, 5> X <1, -9, -14> = <17, -37, 25>