Let $x_0$ be the point of local maxima of $f(x)=\vec{a} \cdot(\vec{b} \times \vec{c})$, where $\vec{a}=x \hat{i}-2 \hat{j}+3 \hat{k}, \vec{b}=-2 \hat{i}+x \hat{j}+\hat{k}$ and $...

The distance of the point $(1,-2,3)$ from the plane $x-y+z=5$ measured parallel to the line $\frac{x}{2}=\frac{y}{3}=\frac{z}{-6}$ is :

Let $\sqrt{3} \hat{i}+\hat{j}, \hat{i}+\sqrt{3} \hat{j}$ and $\beta \hat{i}+(1-\beta) \hat{j}$ respectively be the position vectors of the points $A, B$ and $C$ with respect to the...

If the point $(2, \alpha, \beta)$ lies on the plane which passes through the points $(3,4,2)$ and $(7,0,6)$ and is perpendicular to the plane $...

Let a plane P contain two lines
$$ \overrightarrow{\mathrm{r}}=\hat{\mathrm{i}}+\lambda(\hat{\mathrm{i}}+\hat{\mathrm{j}}), \lambda \in \mathrm{R} \text { and } \overrightarrow{\mathrm{r}}=-\hat{\mathrm{i}}+\mu(\hat{\mathrm{j}}-\hat{\mathrm{k}}), \mu \in \mathrm{R} $$ If $\mathrm{Q}(\alpha, \beta, \gamma)$...

The plane which bisects the line joining, the points (4, –2, 3) and (2, 4, –1) at right angles also passes through the point:

Two lines $\frac{x-3}{1}=\frac{y+1}{3}=\frac{z-3}{4}$ and $\frac{x+5}{7}=\frac{y-2}{-6}=\frac{z-3}{4}$ intersect at the point $R$. The reflection of $R$ in the $x y$-plane has coordinates :

The mirror image of the point $(1,2,3)$ in a plane is $\left(-\frac{7}{3},-\frac{4}{3},-\frac{1}{3}\right)$. Which of the following points lies on this plane?

$f$ the lines $x=a y+b, z=c y+d$ and $x=a^{\prime} z+b^{\prime}, y=c^{\prime} z+d^{\prime}$ are perpendicular, then

Let $\vec{a}=\hat{i}-2 \hat{j}+\hat{k}$ and $\vec{b}=\hat{i}-\hat{j}+\hat{k}$ be two vectors. If $\vec{c}$ is a vector such that $\vec{b} \times \vec{c}=\vec{b} \times \vec{a}$ and $\vec{c} \cdot \vec{a}=0$, then...