Let $\vec{a}$ be a non-zero vector parallel to the line of intersection of the two planes described by $\hat{i}+\hat{j}, \hat{i}+\hat{k}$ and $\hat{i}-\hat{j}, \hat{j}-\hat{k}$. If $\theta$...

Let $(\alpha, \beta, \gamma)$ be the image of the point $\mathrm{P}(2,3,5)$ in the plane $2 \mathrm{x}+\mathrm{y}-3 \mathrm{z}=6$. Then $\alpha+\beta+\gamma$ is equal to

Let $\vec{a}=6 \hat{i}+9 \hat{j}+12 \hat{k}, \vec{b}=\alpha \hat{i}+11 \hat{j}-2 \hat{k}$, and $\vec{c}$ be vectors such that $\vec{a} \times \vec{c}=\vec{a} \times \vec{b}, \vec{a} \cdot \vec{c}=-12, \vec{c}(\hat{i}-2 \hat{j}+\hat{k})=5$,...

Let the plane $P: 4 x-y+z=10$ be rotated by an angle $\frac{\pi}{2}$ about its line of intersection with the plane $x +y-z=4$. If $\alpha$ is...

Let $\lambda \in Z, \vec{a}=\lambda \hat{i}+\hat{j}-\hat{k}$ and $\vec{b}=3 \hat{i}-\hat{j}+2 \hat{k}$. Let $\vec{c}$ be a vector such that $(\vec{a}+\vec{b}+\vec{c}) \times \vec{c}=\overrightarrow{0}, \vec{a} \vec{c}=-17$, $\bar{b} \cdot \bar{c}=-20$....

The shortest distance between the lines $\frac{x-4}{4}=\frac{y+2}{5}=\frac{z+3}{3}$ and $\frac{x-1}{3}=\frac{y-3}{4}=\frac{z-4}{2}$ is -

For any vector $\vec{a}=a_1 \hat{i}+a_2 \hat{j}+a_3 \hat{k}$, with $10\left|a_i\right|<1, i=1,2,3$, consider the following statements
$(\mathrm{A}): \max \left\{\left|\mathrm{a}_1\right|,\left|\mathrm{a}_2\right|,\left|\mathrm{a}_3\right|\right\} \leq|\overrightarrow{\mathrm{a}}|$
$(\mathrm{B}):|\overrightarrow{\mathrm{a}}| \leq 3 \max \left\{\left|\mathrm{a}_1\right|,\left|\mathrm{a}_2\right|,\left|\mathrm{a}_3\right|\right\}$

Let $a, b, c$ be three distinct real numbers, none equal to one. If the vectors $a \hat{i}+\hat{j}+\hat{k}, \hat{i}+b \hat{j}+\hat{k}$ and $\hat{i}+\hat{j}+c \hat{k}$ are coplanar,...

If the points with vectors $\alpha \hat{\mathrm{i}}+10 \hat{\mathrm{j}}+13 \hat{\mathrm{k}}, 6 \hat{\mathrm{i}}+11 \hat{\mathrm{j}}+11 \hat{\mathrm{k}}, \frac{9}{2} \hat{\mathrm{i}}+\beta \hat{\mathrm{j}}-8 \hat{\mathrm{k}}$ are collinear, then $(19 \alpha-6 \beta)^2$ is equal...