The angle made by the position vector of the point $(5, -4, -3)$ with the positive direction of the $X$-axis is:

If the position vectors of three points are $a$,$b$,and $(3a - 2b)$,then these points are .....

$A$ vector $\vec{r}$ has magnitude $14$ and direction ratios $2, 3, -6$. Find the direction cosines and components of $\vec{r}$,given that $\vec{r}$ makes an acute angle with the $x$-axis.

If $\alpha, \beta, \gamma$ are real numbers such that $(\frac{7}{3}+\beta) \hat{i}-\hat{j}+(\alpha+\gamma) \hat{k}=\frac{5}{3}(\alpha \hat{i}+\hat{j}-\hat{k})+\beta(2 \hat{j}+\hat{k})+(\hat{i}+\gamma \hat{j}+3 \hat{k})$, then $5 \alpha-9 \beta+13 \gamma=$

The components of a vector $\vec{a}$ with respect to a rectangular Cartesian system are $2p$ and $1$. The system is rotated about the origin through a certain angle in the counter-clockwise direction. If the components of $\vec{a}$ with respect to the new system are $p + 1$ and $1$,then:

Let $u$ and $v$ be two vectors in a plane. Then any vector $w$ in the plane can be written as $w = au + bv$ for some scalars $a$ and $b$ if and only if