Let $\bar{a} = 2\hat{i} + \hat{j} + \hat{k}$,$\bar{b} = \hat{i} + 2\hat{j} - \hat{k}$,and vector $\bar{c}$ be coplanar with $\bar{a}$ and $\bar{b}$. If $\bar{c}$ is perpendicular to $\bar{a}$,then $\bar{c}$ is:

If $\vec{p}, \vec{q}, \text{ and } \vec{r}$ are three mutually perpendicular vectors of the same magnitude,find the angle between $\vec{p}$ and $\vec{p} + \vec{q} + \vec{r}$.

Let the position vectors of points $A$ and $B$ be $\hat{i}+\hat{j}+\hat{k}$ and $2\hat{i}+\hat{j}+3\hat{k},$ respectively. $A$ point $P$ divides the line segment $AB$ internally in the ratio $\lambda:1$ $(\lambda>0)$. If $O$ is the origin and $\overrightarrow{OB} \cdot \overrightarrow{OP}-3|\overrightarrow{OA} \times \overrightarrow{OP}|^{2}=6,$ then $\lambda$ is equal to

Suppose $\overrightarrow{a}=\lambda \hat{i}-7 \hat{j}+3 \hat{k}$ and $\overrightarrow{b}=\lambda \hat{i}+\hat{j}+2 \lambda \hat{k}$. If the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is greater than $90^{\circ}$, then $\lambda$ satisfies the inequality:

Let the point $A$ divide the line segment joining the points $P(-1, -1, 2)$ and $Q(5, 5, 10)$ internally in the ratio $r : 1$ $(r > 0)$. If $O$ is the origin and $(\overrightarrow{OQ} \cdot \overrightarrow{OA}) - \frac{1}{5}|\overrightarrow{OP} \times \overrightarrow{OA}|^2 = 10$,then the value of $r$ is:

Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|2 \vec{a}+3 \vec{b}|=|3 \vec{a}+\vec{b}|$ and the angle between $\vec{a}$ and $\vec{b}$ is $60^{\circ}$. If $\frac{1}{8} \vec{a}$ is a unit vector,then $|\vec{b}|$ is equal to :