If a particle is acted upon by forces of magnitude $6$ and $7$ units in the directions of $-\hat{i} - 2\hat{j} + 2\hat{k}$ and $2\hat{i} - 3\hat{j} - 6\hat{k}$ respectively,and it undergoes a displacement from point $P(2, -1, -3)$ to $Q(5, -1, 1)$,then the work done by the forces is .......... units.

If $\overrightarrow{a}, \overrightarrow{b}$ and $\overrightarrow{c}$ are unit vectors such that $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=\overrightarrow{0}$,then the value of $3 \overrightarrow{a} \cdot \overrightarrow{b}+2 \overrightarrow{b} \cdot \overrightarrow{c}+\overrightarrow{c} \cdot \overrightarrow{a}$ is

If $\theta$ is the angle between the vectors $\vec{a} = 2\hat{i} + 2\hat{j} - \hat{k}$ and $\vec{b} = 6\hat{i} - 3\hat{j} + 2\hat{k}$,then:

If $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ are three vectors such that $\overrightarrow{a}=\overrightarrow{b}+\overrightarrow{c}$ and the angle between $\overrightarrow{b}$ and $\overrightarrow{c}$ is $\frac{\pi}{2}$, then:

If $\vec{a}$,$\vec{b}$,and $\vec{a}-\vec{b}$ are unit vectors and the angle between the two vectors $\vec{a}$ and $\vec{b}$ is $\theta$,then $\theta = $ . . . . . . .

Let three vectors $\overrightarrow{a}=\alpha \hat{i}+4 \hat{j}+2 \hat{k}$,$\vec{b}=5 \hat{i}+3 \hat{j}+4 \hat{k}$,and $\vec{c}=x \hat{i}+y \hat{j}+z \hat{k}$ form a triangle such that $\overrightarrow{c}=\overrightarrow{a}-\overrightarrow{b}$ and the area of the triangle is $5 \sqrt{6}$. If $\alpha$ is a positive real number,then $|\overrightarrow{c}|^2$ is: