Find the component of vector $\vec{r}$ in the direction of vector $\vec{a}$.

The vectors from the origin to the points $A$ and $B$ are $\overrightarrow A = 3\hat i - 6\hat j + 2\hat k$ and $\overrightarrow B = 2\hat i + \hat j - 2\hat k$ respectively. The area of the triangle $OAB$ is:

If $\vec{A}, \vec{B}$ and $\vec{C}$ are vectors having unit magnitude. If $\vec{A} + \vec{B} + \vec{C} = \vec{0}$,then $\vec{A} \cdot \vec{B} + \vec{B} \cdot \vec{C} + \vec{C} \cdot \vec{A}$ will be:

If $\overrightarrow{A} = a_{1} \hat{\imath} + a_{2} \hat{\jmath}$ and $\overrightarrow{B} = b_{1} \hat{\imath} + b_{2} \hat{\jmath}$ are perpendicular to each other,then:

Vector $\vec{A}$ of magnitude $5 \sqrt{3}$ units,another vector $\vec{B}$ of magnitude $10$ units are inclined to each other at an angle of $30^{\circ}$. The magnitude of the vector product of the two vectors is $\left[\sin 30^{\circ}=\frac{1}{2}\right]$

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 $x$.