Let $\overline{OA} = \vec{a}$,$\overline{OB} = 10\vec{a} + 2\vec{b}$,and $\overline{OC} = \vec{b}$,where $O, A, C$ are non-collinear. Let $p$ be the area of the quadrilateral $OABC$ and $q$ be the area of the parallelogram with adjacent sides $OA$ and $OC$. Then $p/q = \dots$

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

The number of vectors of unit length perpendicular to the two vectors $a=(1,1,0)$ and $b=(0,1,1)$ is

For any vector $x$, where $\hat{i}, \hat{j}, \hat{k}$ have their usual meanings, the value of $(x \times \hat{i})^{2} + (x \times \hat{j})^{2} + (x \times \hat{k})^{2}$ is equal to

Let $\vec{a} = 3\hat{i} + 2\hat{j} + 2\hat{k}$ and $\vec{b} = \hat{i} + 2\hat{j} - 2\hat{k}$ be two vectors. If a vector perpendicular to both the vectors $\vec{a} + \vec{b}$ and $\vec{a} - \vec{b}$ has the magnitude $12$,then one such vector is

Find the unit vector which is perpendicular to the vector $5i + 2j + 6k$ and coplanar with the vectors $2i + j + k$ and $i - j + k$.