If the vectors $\vec P = a\hat i + a\hat j + 3\hat k$ and $\vec Q = a\hat i - 2\hat j - \hat k$ are perpendicular to each other,then the positive value of $a$ is

For any two vectors $\vec{A}$ and $\vec{B}$,if $\vec{A} \cdot \vec{B} = |\vec{A} \times \vec{B}|$,the magnitude of $(\vec{A} + \vec{B})$ is: $(\tan \frac{\pi}{4} = 1, \cos \frac{\pi}{4} = \frac{1}{\sqrt{2}})$

If $\vec{a}$ and $\vec{b}$ make an angle $\cos^{-1}\left(\frac{5}{9}\right)$ with each other,then $|\vec{a}+\vec{b}|=\sqrt{2}|\vec{a}-\vec{b}|$ for $|\vec{a}|=n|\vec{b}|$. The integer value of $n$ is . . . . . . .

If $\overrightarrow{A} = (2\hat{i} + 3\hat{j} - \hat{k}) \; m$ and $\overrightarrow{B} = (\hat{i} + 2\hat{j} + 2\hat{k}) \; m$,the magnitude of the component of vector $\overrightarrow{A}$ along vector $\overrightarrow{B}$ will be $...... \; m$.

If $\overrightarrow{A} \times \overrightarrow{B} = \overrightarrow{B} \times \overrightarrow{A}$,then the angle between $\overrightarrow{A}$ and $\overrightarrow{B}$ is

$A$ vector $\overrightarrow{F}_1$ is along the positive $X$-axis. If its vector product with another vector $\overrightarrow{F}_2$ is zero,then $\overrightarrow{F}_2$ could be: