LinAlg4.2.17

From Exampleproblems

If the two vectors $\bar{a}=2\bar{i}+\bar{j}+2\bar{k},\bar{b}=5\bar{i}-3\bar{j}+\bar{k}\,$ are two vectors,find the projection of $\bar{b}\,$ on $\bar{a}\,$

Orthogonal projection of vector b on vector a is $\frac{(\bar{b}\cdot\bar{a})\bar{a}}{|a|^2}\,$

This is equal to $\frac{[(5\bar{i}-3\bar{j}+\bar{k})\cdot(2\bar{i}+\bar{j}+2\bar{k})](2\bar{i}+\bar{j}+2\bar{k})}{\sqrt{4+1+4}}=\frac{(10-3+2)(2\bar{i}+\bar{j}+2\bar{k})}{9}=2\bar{i}+\bar{j}+2\bar{k}\,$