Let $v$ and $w$ be vectors such that $\rm{proj}_w v = \left(\begin{smallmatrix}4\\-7\end{smallmatrix}\right).$ Find $\rm{proj}_{-2 w} (3 v)$.

Question

Let ${v}$ and ${w}$ be vectors such that $$\text{proj}_{{w}} {v} = \begin{pmatrix} 4 \\ -7 \end{pmatrix}.$$ Find $\text{proj}_{-2 {w}} (3 {v})$.

Answer 1

The equation,

$$\text{proj}_{\vec x} \vec y=\frac{\vec x \cdot \vec y}{|\vec x|} \frac{\vec x}{|\vec x|}$$

Holds for any vectors we can let those be $\pi \vec a$ and $e \vec b$. Where $\pi$ and $e$ are some random constants like $-2$ and $3$.

$$\text{proj}_{\pi \vec a} e \vec b=\frac{ \pi \vec a \cdot e \vec b}{|\pi \vec a|} \frac{\pi \vec a}{|\pi \vec a|}$$

Now use $\vec a \cdot ( c \vec b)=c \vec a \cdot \vec b$ and $|c \vec w|=c |\vec w|$. To get,

$$=\frac{\pi^2 e}{\pi^2} \frac{\vec a \cdot \vec b}{|\vec a|} \frac{\vec a}{|\vec a|}$$

$$=e \left( \text{proj}_{a} b \right)$$