# Find the vector projection and component perpendicular

Question: If vector $$a = <2,3,5>$$ and vector $$b = <2,-2,-1>$$, find the vector projection of b onto a and hence find the component of b perpendicular to a.

I found the vector projection of b onto a using the vector projection formula which is (-7/38)<2,3,5>.

Do I find the component of b perpendicular to a by using vector b - vector projection? Would this be <2,-2,-1> - (-7/38)<2,3,5>?

## 1 Answer

Vector Projection = $$\mbox{proj}_{a}b=\dfrac{a\cdot b}{|a|^2}a=\dfrac{\langle2,3,5\rangle\langle2,-2,-1\rangle}{38}(2)$$

The component of b perpendicular to a is $$b-\mbox{proj}_{a}b$$

• @llamaro25 If you find my answer helpful then accept the answer $\checkmark$ :) – Key Flex Mar 25 at 13:27