The position vectors of the vertices of $\triangle ABC$ are $4\hat{i} - 2\hat{j}$,$\hat{i} + 4\hat{j} - 3\hat{k}$,and $-\hat{i} + 5\hat{j} + \hat{k}$ respectively. Then,$m \angle ABC = $

$A$ hall has a square floor of dimension $10 \, m \times 10 \, m$ and vertical walls. If the angle $GPH$ between the diagonals $AG$ and $BH$ is $\cos^{-1} \frac{1}{5}$,then the height of the hall (in $meters$) is:

Let $\vec{AB} = 2 \hat{i} + 4 \hat{j} - 5 \hat{k}$ and $\vec{AD} = \hat{i} + 2 \hat{j} + \lambda \hat{k}$, $\lambda \in R$. Let the projection of the vector $\vec{v} = \hat{i} + \hat{j} + \hat{k}$ on the diagonal $\vec{AC}$ of the parallelogram $ABCD$ be of length $1$ unit. If $\alpha, \beta$, where $\alpha > \beta$, are the roots of the equation $\lambda^2 x^2 - 6 \lambda x + 5 = 0$, then $2 \alpha - \beta$ is equal to

In a triangle $ABC,$ if $|\overline{BC}|=8, |\overline{CA}|=7, |\overline{AB}|=10,$ then the projection of the vector $\overline{AB}$ on $\overline{AC}$ is equal to ....... .

$A$ unit vector in the plane of the vectors $2i + j + k$ and $i - j + k$ and orthogonal to $5i + 2j + 6k$ is

If vectors $\vec{a}$ and $\vec{b}$ are not perpendicular,and $\vec{c}$ and $\vec{d}$ are two vectors satisfying $\vec{b} \times \vec{c} = \vec{b} \times \vec{d}$ and $\vec{a} \cdot \vec{d} = 0$,then the vector $\vec{d}$ is equal to: