If $l\vec{a} + m\vec{b} + n\vec{c} = \vec{0},$ where $l, m, n$ are scalars and $\vec{a}, \vec{b}, \vec{c}$ are mutually perpendicular non-zero vectors,then

The value of $b$ such that the scalar product of the vector $\vec{a} = \hat{i} + \hat{j} + \hat{k}$ with the unit vector parallel to the sum of the vectors $\vec{u} = 2\hat{i} + 4\hat{j} - 5\hat{k}$ and $\vec{v} = b\hat{i} + 2\hat{j} + 3\hat{k}$ is $1$, is...

If $\overrightarrow{a} \cdot \hat{i} = \overrightarrow{a} \cdot (2 \hat{i} + \hat{j}) = \overrightarrow{a} \cdot (\hat{i} + \hat{j} + 3 \hat{k}) = 1$,then $\overrightarrow{a}$ is equal to :

If $|a+b|^{2}+|a \cdot b|^{2}=144$ and $|a|=6$,then $|b|$ is equal to

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 work done by a force $\vec{F} = 2\hat{i} - \hat{j} - \hat{k}$ on an object moving with a displacement vector $\vec{d} = 3\hat{i} + 2\hat{j} - 5\hat{k}$ is ............ units.