The sum of the lengths of projections of $p \hat{i} + q \hat{j} + r \hat{k}$ on the coordinate axes, where $p = 4, q = -5, r = 7$, is: (in $\text{ units}$)

Let $A$ be the set of vectors $a = (a_1, a_2, a_3)$ satisfying $\left(\sum_{i=1}^3 \frac{a_i}{2^i}\right)^2 = \sum_{i=1}^3 \frac{a_i^2}{2^i}$. 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

If non-zero vectors $\vec{a}$,$\vec{b}$,and $\vec{c}$ are related by $\vec{a} = 8\vec{b}$ and $\vec{c} = -7\vec{b}$,find the angle between $\vec{a}$ and $\vec{c}$.

The vector in the direction of the sum of the vectors $\vec{a}=2 \hat{i}-2 \hat{j}+5 \hat{k}$ and $\vec{b}=-2 \hat{i}+5 \hat{j}-3 \hat{k}$ is

The position vector of point $C$ relative to $B$ is $(\hat{i} + \hat{j})$ and the position vector of $B$ relative to $A$ is $(\hat{i} - \hat{j})$. The position vector of $C$ relative to $A$ is: