Projection of u onto v and v onto u Given the vector u = <−2,6,

stop2dance3l 2021-12-14 Answered
Projection of u onto v and v onto u
Given the vector u=<2,6,4> and a vector v such that the vector projection of u onto v is <2,4,4>, and the vector projection of v onto u is <8,24,16>. What is the vector v?
Let v=<a,b,c>
Projection of v on u is given by:
projuv=u.v|u|2u=u.v(2)2+62+42<2,6,4>
<8,24,16u.v(2)2+62+42<2,6,4>
4<2,6,4u.v4+36+16<2,6,4>
<2,6,4u.v224<2,6,4>
On comparing u.v224=1
u.v=224
projvu=u.v|v|2v=224|v|2<a,b,c>
<2,4,4224|v|2<a,b,c>
Dividing both sides by 2 we get:
<1,2,2112|v|2<a,b,c>
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Expert Answer

stomachdm
Answered 2021-12-15 Author has 33 answers
The projection of u onto v is a scalar times v. So from the given information we have v=λ(2,4,4).
Hence projuv=vuuuu=λ914(2,6,4).
It is given that this projection is (8,24,16),so λ=569 and hence
v=569(2,4,4)

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Karen Robbins
Answered 2021-12-16 Author has 49 answers
Here is a more tangent way to solve this problem:
Notice v and the projection of u onto v must have the same direction, therefore we can assume v=λ(2,4,4),
in which λ is a constant to be determined. Now use the other condition to establish the equation
(8,24,16)=(v,u)(u,u)u=36λ56(2,6,4).
Solve this to get λ=569. Therefore, v=569(2,4,4).

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