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

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
$k ^{2}=\vec{ a }^{2}+\vec{ b }^{2}+\vec{ c }^{2}+2 \vec{ a } \cdot \vec{ b }-2 \vec{ a } \cdot \vec{ c }-2 \vec{ b } \cdot \vec{ c }$
$k ^{2}=\vec{ a }^{2}+\vec{ b }^{2}+\vec{ c }^{2}=36$
$k=6=|\vec{a}+\vec{b}-\vec{c}|$