If the length of a vector is $21$ and its direction ratios are $2, -3, 6$,then its direction cosines are:

If $\hat{i}, \hat{j}, \hat{k}$ are unit vectors,then which of the following is true?

If a vector $a$ of magnitude $50$ is collinear with the vector $b = 6i - 8j - \frac{15}{2}k$, and makes an acute angle with the positive direction of the $z$-axis, then the vector $a$ is equal to:

Let $\vec{a}$ and $\vec{b}$ be two non-collinear vectors. For what values of $x$ and $y$ is the equation $2\vec{u} - \vec{v} = \vec{w}$ true,where $\vec{u} = x\vec{a} + 2y\vec{b}$,$\vec{v} = -2y\vec{a} + 3x\vec{b}$,and $\vec{w} = 4\vec{a} - 2\vec{b}$?

If $\overline{a} = \bar{i} - 2\bar{j} + 2\bar{k}$ and $\overline{b} = 9\bar{i} + 6\bar{j} - 18\bar{k}$ are two vectors, then $\frac{\text{Projection of } \overline{b} \text{ on } \overline{a}}{\text{Projection of } \overline{a} \text{ on } \overline{b}} = $

Find the direction cosines of the vector $\hat{i}+2 \hat{j}+3 \hat{k}$.