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

$\hat{i} \cdot (\hat{k} \times \hat{j}) + \hat{j} \cdot (\hat{i} \times \hat{k}) + \hat{k} \cdot (\hat{i} \times \hat{j}) = \_\_\_\_$

Let $\vec{a}=\hat{i}+2\hat{j}+\hat{k}$ and $\vec{b}=2\hat{i}+\hat{j}-\hat{k}$. Let $\hat{c}$ be a unit vector in the plane of the vectors $\vec{a}$ and $\vec{b}$ and be perpendicular to $\vec{a}$. Then such a vector $\hat{c}$ is :

Let the vectors $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ be such that $|\overrightarrow{a}|=2, |\overrightarrow{b}|=4$ and $|\overrightarrow{c}|=4$. If the projection of $\overrightarrow{b}$ on $\overrightarrow{a}$ is equal to the projection of $\overrightarrow{c}$ on $\overrightarrow{a}$ and $\overrightarrow{b}$ is perpendicular to $\overrightarrow{c}$,then the value of $|\overrightarrow{a}+\overrightarrow{b}-\overrightarrow{c}|$ is

The horizontal force and the force inclined at an angle $60^\circ$ with the vertical,whose resultant is in the vertical direction with a magnitude of $P \ kg$,are:

Let $p=\hat{i}+2\hat{j}-\hat{k}$ and $q=2\hat{i}-\hat{j}+\hat{k}$. If $a$ and $b$ are two vectors such that $p=a-2b$ and $q=2a+b$, then the angle between $a$ and $b$ is