If $\triangle ABC$ is a right-angled triangle in which $BC$ is the hypotenuse, and the position vectors of $B$ and $C$ are $\vec{b} = 3\hat{i} - 2\hat{j} + \hat{k}$ and $\vec{c} = 5\hat{i} + \hat{j} - 3\hat{k}$ respectively, then the value of $\vec{AB} \cdot \vec{AC} + \vec{BA} \cdot \vec{BC} + \vec{CA} \cdot \vec{CB}$ is:

Consider two vectors $\overrightarrow{u} = 3\hat{i} - \hat{j}$ and $\overrightarrow{v} = 2\hat{i} + \hat{j} - \lambda\hat{k}$,where $\lambda > 0$. The angle between them is given by $\cos^{-1}\left(\frac{\sqrt{5}}{2\sqrt{7}}\right)$. Let $\vec{v} = \vec{v}_1 + \vec{v}_2$,where $\vec{v}_1$ is parallel to $\overrightarrow{u}$ and $\vec{v}_2$ is perpendicular to $\overrightarrow{u}$. Then the value $|\vec{v}_1|^2 + |\vec{v}_2|^2$ is equal to

In a plane, $\vec{a}$ and $\vec{b}$ are the position vectors of two points $A$ and $B$ respectively. $A$ point $P$ with position vector $\vec{r}$ moves on that plane in such a way that $|\vec{r}-\vec{a}| - |\vec{r}-\vec{b}| = c$ (where $c$ is a real constant). The locus of $P$ is a conic section whose eccentricity is:

Let $\bar{a} = 2\bar{i} - \bar{j} + \bar{k}$,$\bar{b} = \bar{i} + 2\bar{j} - \bar{k}$,and $\bar{c} = \bar{i} + \bar{j} - 2\bar{k}$ be three vectors. $A$ vector $\bar{r}$ in the plane of $\bar{b}$ and $\bar{c}$ has a projection of magnitude $\sqrt{\frac{2}{3}}$ on the vector $\bar{a}$. Find $\bar{r}$.

If $\vec{a} = 3\hat{i} - 6\hat{j} - 24\hat{k}$,then which of the following vectors is perpendicular to $\vec{a}$?

The angle between the vectors $a + b$ and $a - b$,when $a = (1, 1, 4)$ and $b = (1, -1, 4)$ is .............. $^o$