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

If $\vec{f}, \vec{g}, \vec{h}$ are mutually orthogonal vectors of equal magnitudes,then the angle between the vectors $\vec{f}+\vec{g}+\vec{h}$ and $\vec{h}$ is

Let $\vec{a}$ and $\vec{b}$ be vectors of the same magnitude such that $\frac{|\vec{a}+\vec{b}|+|\vec{a}-\vec{b}|}{|\vec{a}+\vec{b}|-|\vec{a}-\vec{b}|}=\sqrt{2}+1$. Then $\frac{|\vec{a}+\vec{b}|^2}{|\vec{a}|^2}$ is:

Let $\vec{a}=2 \hat{i}+3 \hat{j}+\hat{k}$,$\vec{b}=4 \hat{i}+\hat{j}$,$\vec{c}=\hat{i}-3 \hat{j}-7 \hat{k}$. If $\vec{r}=x \hat{i}+y \hat{j}+z \hat{k}$,$\vec{r} \cdot \vec{a}=9$,$\vec{r} \cdot \vec{b}=7$,$\vec{r} \cdot \vec{c}=6$,then $(x, y, z) = $

If $(\vec{a}+3 \vec{b})$ is perpendicular to $(7 \vec{a}-5 \vec{b})$ and $(\vec{a}-4 \vec{b})$ is perpendicular to $(7 \vec{a}-2 \vec{b})$,then the angle between $\vec{a}$ and $\vec{b}$ (in degrees) is $......$

If $|\vec{a}| = 2\sqrt{2}$,$|\vec{b}| = 3$ and the angle between $\vec{a}$ and $\vec{b}$ is $\frac{\pi}{4}$,find the length of the longer diagonal of the parallelogram whose sides are $5\vec{a} + 2\vec{b}$ and $\vec{a} - 3\vec{b}$.