If $a$ and $b$ are vectors such that $|a+b| = |a-b|$,then the angle between $a$ and $b$ is (in $^{\circ}$)

Find the projection of the vector $\vec{a} = 2\hat{i} + 3\hat{j} + 2\hat{k}$ on the vector $\vec{b} = \hat{i} + 2\hat{j} + \hat{k}$.

If magnitudes of vectors $\vec{a}, \vec{b}, \vec{c}$ are $3, 4,$ and $5$ respectively,and $\vec{a}$ is perpendicular to $\vec{b} + \vec{c}$,$\vec{b}$ is perpendicular to $\vec{c} + \vec{a}$,and $\vec{c}$ is perpendicular to $\vec{a} + \vec{b}$,then find the value of $|\vec{a} + \vec{b} + \vec{c}|$.

If $\hat{a}, \hat{b}$ and $\hat{c}$ are unit vectors such that $\hat{a}+\hat{b}+\hat{c}=\vec{0}$,then the value of $\hat{a} \cdot \hat{b}+\hat{b} \cdot \hat{c}+\hat{c} \cdot \hat{a}$ is equal to

Let $\vec{a} = \hat{i} - 2\hat{j} + 3\hat{k}$. If $\vec{b}$ is a vector such that $\vec{a} \cdot \vec{b} = |\vec{b}|^2$ and $|\vec{a} - \vec{b}| = \sqrt{7}$,then find $|\vec{b}|$.

If $\vec{a}=\hat{i}+\hat{j}+\hat{k}$,$\vec{b}=\hat{i}-\hat{j}+\hat{k}$,and $\vec{c}=\hat{i}-\hat{j}-\hat{k}$ are three vectors,then the vector $\vec{r}$ in the plane of $\vec{a}$ and $\vec{b}$,whose projection on $\vec{c}$ is $\frac{1}{\sqrt{3}}$,is given by: