Let say i have an orthogonal base $B$ which is some base of $R^3$, and now i want to find the vector coordinates of $[1,2,3]$ by base $B$ using the sum of projection vectors.
1.Is there some formula for this calculation? 2. I would like to get the idea behind this question.
Thanks!
Hint:
The scalar projection of a vector $\vec v$ in the direction of another vector $\vec b$ is: $$ v_b=\frac{\vec v \cdot \vec b}{||\vec b||} $$
where ''$\cdot$'' is the dot product.
If you know the vectors $\{\vec b_i\}$ of the basis $B$ and $\vec v=[1,2,3]^T$ is your vector, these scalar projections are the component of $\vec v$ in the basis $B$.