Write $\vec{a}=(3,2,-6)$ as the sum of two vectors, one parallel, and one perpendicular, to vector $\vec{d}=(2,-4,1)$.

Question

The solution is $$\vec{a}=(-8/21)\vec{d}+(79/21,10/21,-118/21)$$

Answer 1

Consider the vector $$ v=a-\frac{a\cdot d}{d\cdot d}d $$ Note that $$ \begin{align} v\cdot d &=a\cdot d-\frac{a\cdot d}{d\cdot d}d\cdot d\\ &=0 \end{align} $$ Therefore, $$ a=v+\frac{a\cdot d}{d\cdot d}d $$ where $v$ is perpendicular to $d$ and $\frac{a\cdot d}{d\cdot d}d$ is parallel to $d$.