If $a(\vec{\alpha} \times \vec{\beta}) + b(\vec{\beta} \times \vec{\gamma}) + c(\vec{\gamma} \times \vec{\alpha}) = \overrightarrow{0}$, where $a, b, c$ are non-zero scalars, then the vectors $\vec{\alpha}, \vec{\beta}, \vec{\gamma}$ are

If $a = t^2 \hat{i} + e^t \hat{j} + \hat{k}$ and $b = 2 \hat{i} + t^2 \hat{j} + \log t \hat{k}$,and $f(t) = a \cdot b$,then $f^{\prime}(1)$ is equal to

The points $A(5, -1, 1)$,$B(7, -4, 7)$,$C(1, -6, 10)$,and $D(-1, -3, 4)$ are vertices of a:

Let $\overline{A}=2 \hat{i}+\hat{k}$,$\overline{B}=\hat{i}+\hat{j}+\hat{k}$ and $\overline{C}=4 \hat{i}-3 \hat{j}+7 \hat{k}$. If a vector $\overline{R}$ satisfies $\overline{R} \times \overline{B}=\overline{C} \times \overline{B}$ and $\overline{R} \cdot \overline{A}=0$,then $\overline{R}$ is given by

Let $\vec{a} \times \vec{b} = 7 \hat{i} - 5 \hat{j} - 4 \hat{k}$ and $\vec{a} = \hat{i} + 3 \hat{j} - 2 \hat{k}$. If the length of the projection of $\vec{b}$ on $\vec{a}$ is $\frac{8}{\sqrt{14}}$,then $|\vec{b}| = $

The projection of $\bar{a} = \hat{i} - 2\hat{j} + \hat{k}$ on $\bar{b} = 2\hat{i} - \hat{j} + \hat{k}$ is