If $a = 2i + 3j - 5k$,$b = mi + nj + 12k$ and $a \times b = 0$,then $(m, n) = $

If $\vec{u}$ and $\vec{v}$ are unit vectors and $\theta$ is the acute angle between them,then $2\vec{u} \times 3\vec{v}$ is a unit vector for

Let $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ be the position vectors of the vertices $A, B, C$ respectively of $\triangle ABC$. The vector area of $\triangle ABC$ is:

Let $\overline{a}=\alpha \hat{i}+3 \hat{j}-\hat{k}$,$\overline{b}=3 \hat{i}-\beta \hat{j}+4 \hat{k}$ and $\overline{c}=\hat{i}+2 \hat{j}-2 \hat{k}$,where $\alpha, \beta \in R$,be three vectors. If the projection of $\overline{a}$ on $\overline{c}$ is $\frac{10}{3}$ and $\overline{b} \times \overline{c}=-6 \hat{i}+10 \hat{j}+7 \hat{k}$,then the value of $\alpha^2+\beta^2-\alpha \beta$ is equal to

Let $\vec{a}, \vec{b}, \vec{c}$ be three vectors such that $|\vec{a}|=\sqrt{31}$,$4|\vec{b}|=|\vec{c}|=2$ and $2(\vec{a} \times \vec{b})=3(\vec{c} \times \vec{a})$. If the angle between $\vec{b}$ and $\vec{c}$ is $\frac{2\pi}{3}$,then $\left(\frac{\vec{a} \times \vec{c}}{\vec{a} \cdot \vec{b}}\right)^2$ is equal to $............$.

If $\vec{a} = \hat{i} + \hat{j} + \hat{k}$,$\vec{a} \cdot \vec{b} = 1$,and $\vec{a} \times \vec{b} = \hat{j} - \hat{k}$,then $\vec{b} = \dots$