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 $(3 \hat{i}+2 \hat{j}-\hat{k})$...

Let ${ }^{\mathrm{a}}=\hat{\mathrm{i}}+\alpha \hat{\mathrm{j}}+3 \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{b}}=3 \hat{\mathrm{i}}+\alpha \hat{\mathrm{j}}+\hat{\mathrm{k}}$. If the area of the parallelogram whose adjacent sides are represented by the vectors $\stackrel{a a}{\mathrm{a}}$...

Let three vectors $\overline{\mathrm{a}}, \overline{\mathrm{b}}$ and $\overline{\mathrm{c}}$ be such that $\overline{\mathrm{a}} \times \overline{\mathrm{b}}=\overline{\mathrm{c}}, \overline{\mathrm{b}} \times-\overline{\mathrm{c}}=\overline{\mathrm{a}}$ and $|\overline{\mathrm{a}}|=2$. Then which one bf the following is not...

Let a vector $\bar{a}$ be coplanar with vectors $\bar{b}=2 \hat{i}+\hat{j}+\hat{k}$ and $\bar{c}=\hat{i}-\hat{j}+\hat{k}$. If $\bar{a}$ is perpendicular to $\bar{d}=3 \hat{i}+2 \hat{j}+6 \hat{k}$, and $|\overline{\mathrm{a}}|=\sqrt{10}$. Then a...

Let $L$ be the line of intersection of planes $\vec{r} \cdot(\hat{i}-\hat{j}+2 \hat{k})=2$ and $\vec{r} \cdot(2 \hat{i}+\hat{j}-\hat{k})=2$. If $P(\alpha, \beta, \gamma)$ is the foot of perpendicular...