Question

Suppose u= <1,-5,4> and, v= <-1,1,1>. Then: The projection of u along v is? The projection of u orthogonal to v is?

Answer 1

`Proj_(v)U=(u\cdot v)/(\left\lVert v \right\rVert^2)v=(-1-5+4)/(\left(√((-1)^2+(1)^2+(1)^2)\right)^2)(-1,1,1)=\left((2)/(3),-(2)/(3),-(2)/(3)\right).`

Explanation:

The orthogonal projection of U along V is a vector defined by
`Proj_(v)u=\lambda v` such that
`v\cdot(u-\lambda v)=0.`. From here we obtain that
`\lambda=(u\cdot v)/(\left\lVert v \right\rVert ^(2)).`