If $\vec{a} = \hat{i} - 2\hat{j} + 2\hat{k}$ and $\vec{b} = 2\hat{i} - 3\hat{j} + \hat{k}$, then the component of $\vec{b}$ perpendicular to $\vec{a}$ is

If the resultant of two forces is of magnitude $P$ and equal to one of them and perpendicular to it,then the other force is

If $i, j, k$ are unit orthonormal vectors and $a$ is a vector,if $a \times r = j$,then $a \cdot r$ is

Let $\vec{a} = \hat{i} + \hat{j} + \hat{k}$, $\vec{b} = \hat{i} - 3\hat{j} + 2\hat{k}$ and $\vec{c} = 3\hat{i} - 2\hat{k}$. If a vector $\vec{p}$ satisfies the conditions $\vec{p} \cdot \vec{c} = 0$ and $\vec{p} \times \vec{a} = \vec{b} \times \vec{a}$, then the value of $|\vec{p}|$ is:

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 $|a \cdot b| = 3$ and $|a \times b| = 4$,then the angle between $a$ and $b$ is