What is the angle between $(\overrightarrow P + \overrightarrow Q)$ and $(\overrightarrow P \times \overrightarrow Q)$?

Find the scalar and vector products of two vectors $\vec{a} = (3 \hat{i} - 4 \hat{j} + 5 \hat{k})$ and $\vec{b} = (-2 \hat{i} + \hat{j} - 3 \hat{k})$.

The resultant of two vectors having magnitudes $2$ and $3$ is $1$. What is their cross product?

Three particles $P, Q$ and $R$ are moving along the vectors $\vec{A}=\hat{i}+\hat{j}, \vec{B}=\hat{j}+\hat{k}$ and $\vec{C}=-\hat{i}+\hat{j}$ respectively. They strike a point and start to move in different directions. Now,particle $P$ is moving normal to the plane containing vectors $\vec{A}$ and $\vec{B}$. Similarly,particle $Q$ is moving normal to the plane containing vectors $\vec{A}$ and $\vec{C}$. The angle between the directions of motion of $P$ and $Q$ is $\cos^{-1}\left(\frac{1}{\sqrt{x}}\right)$. Then the value of $x$ is ...... .

If the vectors $2\hat{i} + 2\hat{j} - 2\hat{k}$,$5\hat{i} + y\hat{j} + \hat{k}$,and $-\hat{i} + 2\hat{j} + 2\hat{k}$ are coplanar,then the value of $y$ is:

If $\overrightarrow {A} = 2\hat i + 3\hat j - \hat k$ and $\overrightarrow {B} = - \hat i + 3\hat j + 4\hat k$,a unit vector perpendicular to both $\overrightarrow {A}$ and $\overrightarrow {B}$ will be