$$x-3=\frac{y-\mathrm{k}}{2}=\mathrm{z}$$ intersect, then the value of $$\mathrm{k}$$ is

If the line $$\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-2}{4}$$ meets the plane $$x+2 y+3 z=15$$ at the point $$P$$, then the distance of $$\mathrm{P}$$ fro...

The equation of line passing through the point $$(1,2,3)$$ and perpendicular to the lines $$\frac{x-2}{3}=\frac{y-1}{2}=\frac{z+1}{-2}$$ and $$\frac{x...

The angle between the line $$\frac{x+1}{2}=\frac{y-2}{1}=\frac{z-3}{-2}$$ and plane $$x-2 y-\lambda z=3$$ is $$\cos ^{-1}\left(\frac{2 \sqrt{2}}{3}\ri...

If the direction cosines $$l, \mathrm{~m}, \mathrm{n}$$ of two lines are connected by relations $$l-5 \mathrm{~m}+3 \mathrm{n}=0$$ and $$7 l^2+5 \math...

The mirror image of the point $$(1,2,3)$$ in a plane is $$\left(-\frac{7}{3},-\frac{4}{3},-\frac{1}{3}\right)$$. Thus, the point _________ lies on thi...

A plane is parallel to two lines, whose direction ratios are $$1,0,-1$$ and $$-1,1,0$$ and it contains the point $$(1,1,1)$$. If it cuts co-ordinate a...

The lines $$\frac{x-1}{3}=\frac{y+1}{2}=\frac{z-1}{5} \quad$$ and $$\frac{x+2}{4}=\frac{y-1}{3}=\frac{z+1}{2}$$

The vector equation of the line $$2 x+4=3 y+1=6 z-3$$ is

The plane through the intersection of planes $$x+y+z=1$$ and $$2 x+3 y-z+4=0$$ and parallel to $$\mathrm{Y}$$-axis also passes through the point...

The perpendicular distance of the origin from the plane $$x-3 y+4 z-6=0$$ is

Two lines $$\frac{x-3}{1}=\frac{y+1}{3}=\frac{z-6}{-1}$$ and $$\frac{x+5}{7}=\frac{y-2}{-6}=\frac{z-3}{4} \quad$$ intersect at the point R. Then refle...

The line $$\frac{x-2}{3}=\frac{y-1}{-5}=\frac{z+2}{2}$$ lies in the plane $$x+3 y-\alpha z+\beta=0$$, then the value of $$\alpha^2+\alpha \beta+\beta^...

Let $$\mathrm{P}$$ be a plane passing through the points $$(2,1,0),(4,1,1)$$ and $$(5,0,1)$$ and $$R$$ be the point $$(2,1,6)$$. Then image of $$R$$ i...

The co-ordinates of the point, where the line through $$A(3,4,1)$$ and $$B(5,1,6)$$ crosses the $$\mathrm{XZ}$$-plane, are

$$\mathrm{ABC}$$ is a triangle in a plane with vertices $$\mathrm{A}(2,3,5), \mathrm{B}(-1,3,2)$$ and $$\mathrm{C}(\lambda, 5, \mu)$$. If median throu...

If a line $$\mathrm{L}$$ is the line of intersection of the planes $$2 x+3 y+z=1$$ and $$x+3 y+2 z=2$$. If line $$\mathrm{L}$$ makes an angle $$\alpha...

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

The co-ordinates of the point, where the line $$\frac{x-1}{2}=\frac{y-2}{-3}=\frac{z+5}{4}$$ meets the plane $$2 x+4 y-\mathrm{z}=3$$, are

The equation of a plane, containing the line of intersection of the planes $$2 x-y-4=0$$ and $$y+2 z-4=0$$ and passing through the point $$(2,1,0)$$, ...

The foot of the perpendicular drawn from the origin to the plane is $$(4,-2,5)$$, then the Cartesian equation of the plane is

A vector $$\overrightarrow{\mathrm{n}}$$ is inclined to $$\mathrm{X}$$-axis at $$45^{\circ}$$, $$\mathrm{Y}$$-axis at $$60^{\circ}$$ and at an acute a...

If the Cartesian equation of a line is $$6 x-2=3 y+1=2 z-2$$, then the vector equation of the line is

The direction cosines $$\ell, \mathrm{m}, \mathrm{n}$$ of the line $$\frac{\mathrm{x}+2}{2}=\frac{2 \mathrm{y}-5}{3} ; \mathrm{z}=-1$$ are

Equation of the plane passing through the point (2, 0, 5) and parallel to the vectors $$\widehat i - \widehat j + \widehat k$$ and $$3\widehat i + 2\w...

The co-ordinates of the point $$\mathrm{P} \equiv(1,2,3)$$ and $$\mathrm{O} \equiv(0,0,0)$$, then the direction cosines of $$\overline{\mathrm{OP}}$$ ...

The equation of the plane containing the line $$\frac{x+1}{-3}=\frac{y-3}{2}=\frac{z+2}{1}$$ and the point $$(0,7,-7)$$ is

The equation of a line passing through $$(3,-1,2)$$ and perpendicular to the lines $$\bar{r}=(\hat{i}+\hat{j}-\hat{k})+\lambda(2 \hat{i}-2 \hat{j}+\ha...

The area of the parallelogram with vertices A(1, 2, 3), B(1, 3, a), C(3, 8, 6) and D(3, 7, 3) is $$\sqrt{265}$$ sq. units, then a =

If the lines $\frac{1-x}{3}=\frac{7 y-14}{2 \lambda}=\frac{z-3}{2}$ and $\frac{7-7 x}{3 \lambda}=\frac{y-5}{1}=\frac{6-z}{5}$ are at right angles, the...

The Cartesian equation of the plane passing through the point A(7, 8, 6) and parallel to the XY plane is

The equation of the plane passing through $$(-2,2,2)$$ and $$(2,-2,-2)$$ and perpendicular to the plane $$9 x-13 y-3 z=0$$ is

The Cartesian equation of the line passing through the points A(2, 2, 1) and B(1, 3, 0) is

The Cartesian equation of the plane $$\overline{\mathrm{r}}=(\hat{\mathrm{i}}-\hat{\mathrm{j}})+\lambda(\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm...

The equation of the plane that contains the line of intersection of the planes. $$x+2 y+3 z-4=0$$ and $$2 x+y-z+5=0$$ and is perpendicular to the plan...

The vector equation of the line whose Cartesian equations are y = 2 and 4x $$-$$ 3z + 5 = 0 is

The Cartesian equation of the plane passing through the point $$(0,7,-7)$$ and containing the line $$\frac{x+1}{-3}=\frac{y-3}{2}=\frac{z+2}{1}$$ is...

If the lines $$\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}$$ and $$\frac{x-3}{1}=\frac{y-k}{2}=\frac{z}{1}$$ intersect, then the values of $$k$$ is...

The parametric equations of a line passing through the points $$\mathrm{A}(3,4,-7)$$ and $$\mathrm{B}(1,-1,6)$$ are

The angle between a line with direction ratios 2, 2, 1 and a line joining (3, 1, 4) and (7, 2, 12) is

If the line $$\frac{x+1}{2}=\frac{y-m}{3}=\frac{z-4}{6}$$ lies in the plane $$3 x-14 y+6 z+49=0$$, then the value of $$m$$ is