Let the plane passing through the point (–1, 0, –2) and perpendicular to each of the planes 2x + y – z = 2 and...

Let $L$ be a line obtained from the intersection of two planes $x+2 y+z=6$ and $y+2 z=4$. If point $P(\alpha, \beta, \gamma)$ is the foot...

If $|\vec{a}|=2, \mid \vec{b}=5$ and $|\vec{a} \times \vec{b}|=8$, then $|\vec{a} \cdot \vec{b}|$ is equal to:

Let $a, b$ and $c$ be distinct positive numbers. If the vectors $a-a \hat{i}+a \hat{j}+c \hat{k}, \hat{i}+\hat{k}$ and $c \hat{i}+c \hat{j}+b \hat{k}$ are co-planar, then...

If the shortest distance between the straight lines $3(x-1)=6(y-2)=2(z-1)$ and $4(x-2)=2(y-\lambda)=(z-3)$, $\lambda \in \mathrm{R}$ is $\frac{1}{\sqrt{38}}$, then the integral value of $\lambda$ is equal to:

Let $\vec{a}=\hat{i}+\hat{j}+2 \hat{k}$ and $\vec{b}=-\hat{i}+2 \hat{j}+3 \hat{k}$. Then the vector product $(\vec{a}+\vec{b}) \times((\vec{a} \times((\vec{a}-\vec{b}) \times \vec{b})) \times \vec{b})$ is equal to.

If the shortest distance between the lines $\vec{r}_1=\alpha \hat{i}+2 \hat{j}+2 \hat{k}+\lambda(\hat{i}-2 \hat{j}+2 \hat{k}), \lambda \in R, \alpha>0$ and $...

Let $P$ be a plane passing through the points $(1,0,1),(1,-2,1)$ and $(0,1,-2)$. Let a vector $\overrightarrow{\mathbf{a}}=\alpha \hat{\mathbf{i}}+\beta \hat{\mathbf{j}}+\gamma \hat{\mathbf{k}}$ be such that $\vec{a}$ is parallel...

Let $\vec{a}, \vec{b}, \vec{c}$-be three mutually perpendicular vectors of the same magnitude and equally inclined at an angle $\theta$, with the vector $\vec{a}+\vec{b}+\vec{c}$. Then $...

Let $\vec{a}-=2 \hat{i}+\hat{j}-2 \hat{k}$ and $\vec{b}-=\hat{i}+\hat{j}$. If $\vec{c}-$ is a vector such that $\vec{a} \cdot \vec{c}=|\vec{c}|,|\vec{c}-\vec{a}|=2 \sqrt{2}$ and the angle between $(\vec{a} \times \vec{b})$ and...