Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{a}|=\sqrt{14}$,$|\vec{b}|=\sqrt{6}$ and $|\vec{a} \times \vec{b}|=\sqrt{48}$. Then $(\vec{a} \cdot \vec{b})^2$ is equal to $...........$.

The angle between two diagonals of a cube is:

Three vectors $\vec{a}, \vec{b}, \vec{c}$ satisfy the condition $\vec{a}+\vec{b}+\vec{c}=\vec{0}$. If $|\vec{a}|=1, |\vec{b}|=3, |\vec{c}|=4$,then find the value of $\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}$.

If $a, b, c$ are vectors of equal magnitude such that $(a, b)=\alpha, (b, c)=\beta, (c, a)=\gamma$,then the minimum value of $\cos \alpha+\cos \beta+\cos \gamma$ is

Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three vectors such that $|\vec{a}|=3, |\vec{b}|=4, |\vec{c}|=5$ and each one of them is perpendicular to the sum of the other two. Find $|\vec{a}+\vec{b}+\vec{c}|$.

If $a = 4 \hat{i} + 6 \hat{j}$,$b = 3 \hat{j} + 4 \hat{k}$,and $c$ is the projection vector of $a$ on $b$,then $c$ and $|c|$ respectively are