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

If $a, b, c, d$ are the position vectors of the points $A, B, C$ and $D$ respectively,referred to the same origin $O$,such that no three of these points are collinear and $a + c = b + d$,then the quadrilateral $ABCD$ is a

Let $\overline{a}=2 \hat{i}+\hat{j}-2 \hat{k}$ and $\overline{b}=\hat{i}+\hat{j}$. Let $\overline{c}$ be a vector such that $|\bar{c}-\bar{a}|=3$ and $|(\bar{a} \times \bar{b}) \times \bar{c}|=3$ and the angle between $\overline{c}$ and $\overline{a} \times \overline{b}$ is $30^{\circ}$,then $\overline{a} \cdot \overline{c}$ is equal to

Three vectors $\vec{a}, \vec{b}$ and $\vec{c}$ satisfy $\vec{a}+\vec{b}+\vec{c}=\vec{0}$. If $|\vec{a}|=3, |\vec{b}|=4, |\vec{c}|=2$, then $\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}+2(|\vec{a}|+|\vec{b}|+|\vec{c}|)=$

The orthogonal projection vector of $\vec{a} = 2\hat{i} + 3\hat{j} + 3\hat{k}$ on $\vec{b} = \hat{i} - 2\hat{j} + \hat{k}$ is

If $a, b$ and $c$ are three vectors such that $|a|=3, |b|=4$ and $|c|=5$ and $a+b+c=0$,then $a \cdot b$ is equal to