Let $|\vec{a}|=2, |\vec{b}|=3$ and the angle between $\vec{a}$ and $\vec{b}$ be $\frac{\pi}{3}$. If a parallelogram is constructed with adjacent sides $2\vec{a}+3\vec{b}$ and $\vec{a}-\vec{b}$,then its shorter diagonal is of length

If the coordinates of the points $P$ and $Q$ are $(1, -2, 1)$ and $(2, 3, 4)$ respectively,and $O$ is the origin $(0, 0, 0)$,then which of the following is true?

$A$ unit vector in the $xy$-plane which is perpendicular to $4i - 3j + k$ is

$a$ and $b$ are non-collinear vectors,$|a|=2 \sqrt{2}$,$|b|=3$ and the angle between $a$ and $b$ is $45^{\circ}$. Then,the lengths of the diagonals of the parallelogram whose adjacent sides are represented by the vectors $5a+2b$ and $a-3b$ are

If $\bar{a}$ and $\bar{b}$ are not perpendicular to each other,$\bar{r} \times \bar{a} = \bar{b} \times \bar{a}$ and $\bar{r} \cdot \bar{c} = 0$,then $\bar{r} =$

Suppose $\overrightarrow{a}=\lambda \hat{i}-7 \hat{j}+3 \hat{k}$ and $\overrightarrow{b}=\lambda \hat{i}+\hat{j}+2 \lambda \hat{k}$. If the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is greater than $90^{\circ}$,then $\lambda$ satisfies the inequality: