# Vector Manipulation (Orthogonal and Parallel Vectors)

Consider the vectors a=<2,4,-3> and b=<4,-5,6>. Determine vectors v and w such that a=v+w and v is parallel to b while w is orthogonal to b.

The dot product of two orthogonal vectors is zero and the cross product of two parallel vectors is zero. A parallel vector is a multiple of the chosen vector.

I tried using multiples of b for v and then seeing if random vectors orthogonal to b can be added to v to give a. I'm lost here.

Related Calculus and Beyond Homework Help News on Phys.org
Dick
Homework Helper
Consider the vectors a=<2,4,-3> and b=<4,-5,6>. Determine vectors v and w such that a=v+w and v is parallel to b while w is orthogonal to b.

The dot product of two orthogonal vectors is zero and the cross product of two parallel vectors is zero. A parallel vector is a multiple of the chosen vector.

I tried using multiples of b for v and then seeing if random vectors orthogonal to b can be added to v to give a. I'm lost here.
Instead of expressing parallel as a cross product, if v is parallel to b then v must be a scalar multiple of b. So v=tb for some t. That means w=a-v=a-tb. Now w.b must be 0. Try to solve for t.

Instead of expressing parallel as a cross product, if v is parallel to b then v must be a scalar multiple of b. So v=tb for some t. That means w=a-v=a-tb. Now w.b must be 0. Try to solve for t.
I tried using w=a-tb. I dotted both sides by b, and got 0=(a-tb).b, where t is the only unknown, but I got stuck again. I'm not quite sure how to solve for t at that point.

Dick