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Vector Manipulation (Orthogonal and Parallel Vectors)

  • Thread starter adam199
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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.
 

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  • #2
Dick
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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.
 
  • #3
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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.
 
  • #4
Dick
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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.
Distribute the dot product. (a-tb).b=a.b-t(b.b)=0. Now try it.
 
  • #5
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Distribute the dot product. (a-tb).b=a.b-t(b.b)=0. Now try it.
thanks
 

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