If $i, j, k$ are unit orthonormal vectors and $a$ is a vector,if $a \times r = j$,then $a \cdot r$ is

If $\vec{\alpha} = 3\hat{i} - \hat{k}$, $|\vec{\beta}| = \sqrt{5}$, and $\vec{\alpha} \cdot \vec{\beta} = 3$, then the area of the parallelogram for which $\vec{\alpha}$ and $\vec{\beta}$ are adjacent sides is:

If $a, b, c$ are vectors of equal magnitude such that $(a, b)=\alpha, (b, c)=\beta, (c, a)=\gamma$,then the minimum value of $\cos \alpha+\cos \beta+\cos \gamma$ is

Find the projection of the vector $\hat{i}+3 \hat{j}+7 \hat{k}$ on the vector $7 \hat{i}-\hat{j}+8 \hat{k}$.

If $|\vec{a}|=4$ and $|\vec{b}|=5$,then the values of $k$ for which $\vec{a}+k \vec{b}$ is perpendicular to $\vec{a}-k \vec{b}$ are

Let the vectors $\vec{a} = -\hat{i} + \hat{j} + 3\hat{k}$ and $\vec{b} = \hat{i} + 3\hat{j} + \hat{k}$. For some $\lambda, \mu \in \mathbb{R}$, let $\vec{c} = \lambda \vec{a} + \mu \vec{b}$. If $\vec{c} \cdot (3\hat{i} - 6\hat{j} + 2\hat{k}) = 10$ and $\vec{c} \cdot (\hat{i} + \hat{j} + \hat{k}) = -2$, then $|\vec{c}|^2$ is equal to: