If $\overrightarrow A = 3\hat i + \hat j + 2\hat k$ and $\overrightarrow B = 2\hat i - 2\hat j + 4\hat k$,then find $|\overrightarrow A \times \overrightarrow B |$.

Vectors $a\hat{i} + b\hat{j} + \hat{k}$ and $2\hat{i} - 3\hat{j} + 4\hat{k}$ are perpendicular to each other. Given that $3a + 2b = 7$, and the ratio of $a$ to $b$ is $x/2$, find the value of $x$.

The angle between force $\vec{F}=3 \hat{i}+4 \hat{j}-5 \hat{k}$ and displacement $\vec{d}=5 \hat{i}+4 \hat{j}+3 \hat{k}$ is

If $\overrightarrow{A} = 2\widehat{i} - 2\widehat{j}$ and $\overrightarrow{B} = 2\widehat{k}$,then find the dot product $\overrightarrow{A} \cdot \overrightarrow{B}$.

If $\overrightarrow A = 3\hat i + \hat j + 2\hat k$ and $\overrightarrow B = 2\hat i - 2\hat j + 4\hat k$,then the value of $|\overrightarrow A \times \overrightarrow B |$ will be:

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 . . . . . . .