If $\overline{e_1}, \overline{e_2}$ are two non-collinear unit vectors such that $|\overline{e_1}+\overline{e_2}|=\sqrt{3}$,then $(2 \overline{e_1}-5 \overline{e_2}) \cdot (3 \overline{e_1}+\overline{e_2}) = $

If $|\vec{a}| = \sqrt{27}$,$|\vec{b}| = 7$ and $|\vec{a} \times \vec{b}| = 35$,then $\vec{a} \cdot \vec{b}$ is equal to

Let $\vec{a} = \hat{i} + \hat{j} + \hat{k}$,$\vec{b} = \hat{i} - \hat{j} + \hat{k}$,and $\vec{c} = \hat{i} + \hat{j} - \hat{k}$ be three vectors. If $\vec{v}$ is a vector in the plane of $\vec{a}$ and $\vec{b}$ such that the projection of $\vec{v}$ on $\vec{c}$ is $\frac{1}{\sqrt{3}}$,then $\vec{v} = $

Find the projection of the vector $\hat{i}-\hat{j}$ on the vector $\hat{i}+\hat{j}$.

In a $\triangle ABC$,$|CB|=a$,$|CA|=b$,$|AB|=c$ and $CD$ is the median through the vertex $C$. Then,$CA \cdot CD=$

If the vectors $a\,i - 2j + 3k$ and $3i + 6j - 5k$ are perpendicular to each other,then $a$ is given by