[Math] Prove the formula for direction cosines.

Question

Pf.

END pf.

I have no idea where they came up with the fact that $a=\hat{i}* \hat{v}$ $/$ $|i|$ or $b=\hat{j}* \hat{v}$ $/$ $|j|$ etc. Can I can get proof explanation?

I don't see this equality:

Answer 1

Note that $\hat{i}$, $\hat{j}$ and $\hat{k}$ are orthonormal. Consequently, $\hat{i}\cdot\hat{j}=\hat{i}\cdot\hat{k}=\hat{j}\cdot\hat{k}=0$, and $\hat{i}\cdot\hat{i}=\hat{j}\cdot\hat{j}=\hat{k}\cdot\hat{k}=1$. Therefore, if $\mathbf{v} = a\hat{i}+b\hat{j}+c\hat{k}$ then $\hat{i}\cdot\mathbf{v} = a \hat{i}\cdot\hat{i}+ b \hat{i}\cdot\hat{j} + c \hat{i}\cdot\hat{k}=a$. Similarly, $\hat{j}\cdot\mathbf{v}=b$, and $\hat{k}\cdot\mathbf{v}=c$.