Question

Let the vectors $\vec{a},\vec{b},\vec{c}$ be such that $|\vec{a}|=2,|\vec{b}|=4$ and $|\vec{c}|=4$. If the projection of $\vec{b}$ on $\vec{a}$ is equal to the projection of $\vec{c}$ on $\vec{a}$ and $\vec{b}$ is perpendicular to $\vec{c},$ then the value of $\mid \vec{a}+\vec{b}-\vec{c}$ is ______

Answer 1

Answer: 6.00

Projection of $\vec{b}$ on $\vec{a}=$ projection of $\vec{c}$ on $\vec{a}$
$\Rightarrow \frac{\vec{b}\cdot \vec{a}}{|\vec{a}|}=\frac{\vec{c}\cdot \vec{a}}{|\vec{a}|}\Rightarrow \vec{b}\cdot \vec{a}=\vec{c}\cdot \vec{a}$
$\because \vec{b}$ is perpendicular to $\vec{c}$
$\Rightarrow \vec{b}\cdot \vec{c}=0$
Let $|\vec{a}+\vec{b}-\vec{c}|=k$
Square both sides
$k^2={\vec{a}}^2+{\vec{b}}^2+{\vec{c}}^2+2\vec{a}\cdot \vec{b}-2\vec{a}\cdot \vec{c}-2\vec{b}\cdot \vec{c}$
$k^2={\vec{a}}^2+{\vec{b}}^2+{\vec{c}}^2=36$
$k=6=|\vec{a}+\vec{b}-\vec{c}|$