If $3\vec{a} - 5\vec{b}$ and $2\vec{a} + \vec{b}$ are perpendicular to each other,and $\vec{a} + 4\vec{b}$ and $-\vec{a} + \vec{b}$ are also perpendicular to each other,and $\theta$ is the angle between $\vec{a}$ and $\vec{b}$,then find $\cos \theta$.

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 ............... $unit$.

Let $\vec{u} = 2 \hat{i} + \hat{j}$ and $\vec{v} = 3 \hat{i} - 5 \hat{j}$. Consider three points $P, Q,$ and $R$ having the position vectors $\left(\frac{5}{2}\right) \hat{i} - 2 \hat{j}, \left(\frac{7}{3}\right) \hat{i} - \hat{j},$ and $\left(\frac{9}{4}\right) \hat{i}$ respectively. Among these,the points on the line passing through $\vec{u}$ and $\vec{v}$ are

If $12 \hat{i}-12 \hat{j}-18 \hat{k}$,$-3 \hat{i}-6 \hat{j}-9 \hat{k}$ and $3 \hat{i}+3 \hat{j}-24 \hat{k}$ are the position vectors of the vertices $A, B$ and $C$ respectively of $\triangle ABC$,then the position vector of the incentre of $\triangle ABC$ is

Let $\overline{u}, \overline{v}, \overline{w}$ be vectors such that $|\overline{u}|=1, |\overline{v}|=2, |\overline{w}|=3$. If the projection of $\overline{v}$ along $\overline{u}$ is equal to that of $\overline{w}$ along $\overline{u}$,and the vectors $\overline{v}$ and $\overline{w}$ are perpendicular to each other,then $|\overline{u}-\overline{v}+\overline{w}|$ equals:

If $a, b$ and $c$ are non-coplanar unit vectors such that the angle between any two of them is $60^\circ$, and the vector $d = xa + yb + zc$ is perpendicular to both $a$ and $b$, then the value of $\frac{(x + y)}{z}$ is