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...

Let P be an arbitrary point having sum of the squares of the distance from the planes $\mathrm{x}+\mathrm{y}+\mathrm{z}=0, l \mathrm{x}-\mathrm{nz}=0$ and $\mathrm{x}-2 \mathrm{y}+\mathrm{z}=0$, equal to...

Let $\vec{x}$ be a vector in the plane containing vectors $\vec{a}=2 \hat{i}-\hat{j}+\hat{k}$ and $\vec{b}=\hat{i}+2 \hat{j}-\hat{k}$. If the vector $\vec{x}$ is perpendicular to $(3 \hat{i}+2 \hat{j}-\hat{k})$...