Need a little help with these vectors

Question

Let $v_1$, $v_2$, $v_3$ be mutually orthogonal non-zero vectors in $3$-space. So, any vector $v$ can be expressed as $v = c_1 v_1 + c_2 v_2 + c_3 v_3$.

(a)Show that the scalars $c_1$, $c_2$, $c_3$ are given by the formula $\displaystyle c_i=\frac{v\cdot v_i}{||v_i||^2}$, $i=1,2,3$

how to calculate the value of $v\cdot v_i$? what distributivity of scalar multiplication and addition?

Answer 1

Given $v = c_1v_1 + c_2v_2 + c_3v_3$, take the dot product of both sides with $v_3$. The result is $v\cdot v_3 = c_3||v_3||^2$. Solve for $c_3$. Likewise for $i = 1, 2$.

Answer 2

For i=1

$c_1 =\frac{vv_i}{|v_i|^2} \Leftrightarrow c_1 =\frac{(c_1v_1 + c_2v_2 + c_3v_3)v_i}{|v_i|^2} \Leftrightarrow c_1 =\frac{c_1v_1^2}{|v_i|^2} $ which is true by the fact $v_1^2=|v_1|^2$