If $U = [2, -3, 4]$,$X = [0, 2, 3]$,$V = \begin{bmatrix} 3 \\ 2 \\ 1 \end{bmatrix}$ and $Y = \begin{bmatrix} 2 \\ 2 \\ 4 \end{bmatrix}$,then $UV + XY = $?

If $a \neq 0, b \neq 0$ and $|a + b| = |a - b|,$ then the vectors $a$ and $b$ are

If the vectors $\hat{i}-2x\hat{j}-3y\hat{k}$ and $\hat{i}+3x\hat{j}+2y\hat{k}$ are orthogonal to each other,then the locus of the point $(x, y)$ is

If $D$ and $E$ are the midpoints of the sides $BA$ and $BC$ of triangle $ABC$, then $AE + DC =$

Let $\vec{u} = 2 \hat{i} + \hat{j}$ and $\vec{v} = 3 \hat{i} - 5 \hat{j}$. Consider three points $P, Q,$ and $R$ having the position vectors $\left(\frac{5}{2}\right) \hat{i} - 2 \hat{j}, \left(\frac{7}{3}\right) \hat{i} - \hat{j},$ and $\left(\frac{9}{4}\right) \hat{i}$ respectively. Among these,the points on the line passing through $\vec{u}$ and $\vec{v}$ are

Let $u, v, w$ be vectors such that $|u| = 1, |v| = 2, |w| = 3$. If the projection of $v$ along $u$ is equal to the projection of $w$ along $u$,and $v$ and $w$ are perpendicular to each other,then $|u - v + w|$ equals: