If $\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0}$, $|\vec{a}|=3$, $|\vec{b}|=5$, and $|\vec{c}|=7$, then the angle between $\vec{a}$ and $\vec{b}$ is

If $\vec{a}$ and $\vec{b}$ are two vectors such that $\frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|} < 0$ and $|\vec{a} \cdot \vec{b}|=|\vec{a} \times \vec{b}|$,then the angle between the vectors $\vec{a}$ and $\vec{b}$ is

Three vectors $\vec{a}, \vec{b}$ and $\vec{c}$ satisfy the condition $\vec{a}+\vec{b}+\vec{c}=\vec{0}.$ Evaluate the quantity $\mu=\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a},$ if $|\vec{a}|=1, |\vec{b}|=4$ and $|\vec{c}|=2.$

If $\vec{a}=2 \hat{i}+\hat{j}-\hat{k}$, $\vec{b}=\hat{i}-\hat{j}+3 \hat{k}$, $\vec{x}=\left(\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2}\right) \vec{b}$, $\vec{y}=\left(\frac{\vec{a} \cdot \vec{b}}{|\vec{a}|^2}\right) \vec{a}$ and $\theta$ is the angle between $\vec{a}$ and $\vec{b}$, then $x^2+y^2=$

If $a \neq 0, b \neq 0$ and $|a + b| = |a - b|$,then the vectors $a$ and $b$ are . . . .

Find $|\vec{x}|$,if for a unit vector $\vec{a}$,$(\vec{x}-\vec{a}) \cdot (\vec{x}+\vec{a}) = 12$.