In the above figure,$P$ divides $AC$ in the ratio $3:4$ and $Q$ divides $BC$ in the ratio $4:3$. Then $M$ divides $AQ$ in the ratio:

If $\overline{a}=2 \hat{i}+2 \hat{j}+3 \hat{k}$,$\overline{b}=-\hat{i}+2 \hat{j}+\hat{k}$ and $\overline{c}=3 \hat{i}+\hat{j}$ such that $\overline{b}+\lambda \overline{a}$ is perpendicular to $\overline{c}$,then $\lambda$ is

If $a \cdot b = 0$ and $a + b$ makes an angle $60^{\circ}$ with $a$,then

If $\vec{a} = 2 \hat{i} - \hat{j} + \hat{k}$,$\vec{b} = \hat{i} + \hat{j} - 2 \hat{k}$,and $\vec{c} = \hat{i} + 3 \hat{j} - \hat{k}$ are given vectors. If $\vec{a}$ is perpendicular to $\lambda \vec{b} + \vec{c}$,then $\lambda = . . . . . .$.

The angle between the vectors $\bar{a} = 6 \hat{i} + 2 \hat{j} - 8 \hat{k}$ and $\bar{b} = 4 \hat{i} - 4 \hat{j} + 2 \hat{k}$ is . . . . . . .

Let $a, b$ and $c$ be vectors with magnitudes $3, 4$ and $5$ respectively and $a + b + c = 0$. Then the value of $a \cdot b + b \cdot c + c \cdot a$ is: