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 $\vec{p} + \vec{q} + \vec{r}$.

If $\theta$ is the angle between two vectors $\vec{a}$ and $\vec{b},$ then $\vec{a} \cdot \vec{b} \ge 0$ if

The value of $b$ such that the scalar product of the vector $(i + j + k)$ with the unit vector parallel to the sum of the vectors $(2i + 4j - 5k)$ and $(bi + 2j + 3k)$ is $1$,is:

Let $\vec{a}, \vec{b}, \vec{c}, \vec{d}$ be four vectors such that $\vec{a}$ is perpendicular only to $\vec{c}$. If the vector $\vec{b}$ is parallel to $(\vec{c}-\vec{d})$,then $\vec{c}$ is equal to:

Let two non-collinear unit vectors $\hat{a}$ and $\hat{b}$ form an acute angle. $A$ point $P$ moves,so that at any time $t$ the position vector $\overline{OP}$,where $O$ is the origin,is given by $\hat{a} \cos t + \hat{b} \sin t$. When $P$ is farthest from origin $O$,let $M$ be the length of $\overline{OP}$ and $\hat{u}$ be the unit vector along $\overline{OP}$,then

The angle between unit vectors $\bar{a}$ and $\bar{b}$ in $\mathbb{R}^3$ is $\theta$. Then,the value of $\left|\frac{\bar{a} \cdot \bar{a}}{\bar{a} \cdot \bar{b}} \cdot \frac{\bar{b} \cdot \bar{a}}{\bar{b} \cdot \bar{b}}\right| + |\bar{a} \times \bar{b}|^2$ is: