The position vector of a point $C$ with respect to $B$ is $i + j$ and that of $B$ with respect to $A$ is $i - j$. The position vector of $C$ with respect to $A$ is

If three points $A$,$B$,and $C$ have position vectors $(1, x, 3)$,$(3, 4, 7)$,and $(y, -2, -5)$ respectively and if they are collinear,then $(x, y)$ is

Vectors $\vec{a}$,$\vec{b}$,and $\vec{c}$ are of the same length and they make equal angles with each other when taken in pairs. If $\vec{a} = \hat{i} - \hat{j}$,$\vec{b} = \hat{j} + \hat{k}$,and $\vec{c}$ makes an obtuse angle with the $x$-axis,find the vector $\vec{c}$.

If $A(2 \hat{i} + \hat{j} - \hat{k})$,$B(\lambda \hat{i} + 5 \hat{j} + 4 \hat{k})$,$C(-4 \hat{i} + 3 \hat{j} + 2 \hat{k})$ and $D(-\hat{i} - 2 \hat{j} + 3 \hat{k})$ are four points in space such that $\overrightarrow{AB} = x \overrightarrow{AC} + y \overrightarrow{AD}$ for some real numbers $x \neq 0, y \neq 0$,then $17(\lambda + 9) =$ ?

The direction cosine of the vector $\vec{a} = 3\hat{i} + 4\hat{j} + 5\hat{k}$ in the direction of the positive $x$-axis is:

The position vector of a point that divides the line segment joining $P \equiv(1,2,-1)$ and $Q \equiv(-1,1,1)$ externally in the ratio $1: 2$ is: