If $|a| = 3$,$|b| = 4$ and the angle between $a$ and $b$ is $120^\circ$,then $|4a + 3b| = $

Forces $3i + 2j + 5k$ and $2i + j - 3k$ are acting on a particle and displace it from the point $2i - j - 3k$ to the point $4i - 3j + 7k$. The work done by the forces is ............... $unit$.

If $\vec{l}, \vec{m}, \vec{n}$ are coplanar vectors,for what value of $\lambda$ are the points with position vectors $\vec{l} - 2\vec{m} + 3\vec{n}$,$2\vec{l} + \lambda\vec{m} - 4\vec{n}$,and $-7\vec{m} + 10\vec{n}$ collinear?

If $x$ and $y$ are two unit vectors and $\theta$ is the angle between them,then $\frac{1}{2}|x-y|$ is equal to

If $|\vec{a}| = 3, |\vec{b}| = 4, |\vec{c}| = 5$ such that each vector is perpendicular to the sum of the other two, then $|\vec{a} + \vec{b} + \vec{c}|$ is equal to

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