Let $\left|\overrightarrow{A_1}\right|=3,\left|\overrightarrow{A_2}\right|=5$ and $\left|\overrightarrow{A_1}+\overrightarrow{A_2}\right|=5$. The value of $\left(2 \overrightarrow{A_1}+3 \overrightarrow{A_2}\right) \cdot\left(3 \overrightarrow{A_1}-2 \overrightarrow{A_2}\right)$ is:

If $\vec{A}$ and $\vec{B}$ are two vectors satisfying the relation $\vec{A} \cdot \vec{B}=|\vec{A} \times \vec{B}|$. Then the value of $|\vec{A}-\vec{B}|$ will be :

Two vectors $\vec{P}$ and $\vec{Q}$ have equal magnitudes. If the magnitude of $\vec{P}+\vec{Q}$ is $n$ times magnitude of $\vec{P}-\vec{Q}$, then angle between $\vec{P}$ and $\vec{Q}$...

In an octagon ABCDEFGH of equal side, what is the sum of
$\overrightarrow{\mathrm{AB}}+\overrightarrow{\mathrm{AC}}+\overrightarrow{\mathrm{AD}}+\overrightarrow{\mathrm{AE}}+\overrightarrow{\mathrm{AF}}+\overrightarrow{\mathrm{AG}}+\overrightarrow{\mathrm{AH}}$
if, $\overrightarrow{A O}=2 \hat{i}+3 \hat{j}-4 \hat{k}$

If $\vec{A}=(2 \hat{i}+3 \hat{j}-\hat{k}) \mathrm{m}$ and $\vec{B}=(\hat{i}+2 \hat{j}+2 \hat{k}) \mathrm{m}$. The magnitude of component of vector $\vec{A}$ along vector $\vec{B}$ will be $\_\_\_\_$ m.

If the projection of $2 \hat{i}+4 \hat{j}-2 \hat{k}$ on $\hat{i}+2 \hat{j}+\alpha \hat{k}$ is zero. Then, the value of $\alpha$ will be

Two vectors $\vec{A}$ and $\vec{B}$ have equal magnitudes. If-magnitude of $\vec{A}+\vec{B}$ is equal to two times the magnitude of $\vec{A}-\vec{B}$, then the angle between $\vec{A}$...

If force $\vec{F}=3 \hat{i}+4 \hat{j}-2 \hat{k}$ acts on a particle having position vector $2 \hat{i}+\hat{j}+2 \hat{k}$ then, the torque about the origin will be :-