Three Dimensional Geometry

The angle between the lines whose direction ratios are $a, b, c$ and $b - c, c - a, a - b$ is

The measure of the angle between the lines $x = k + 1, y = 2k - 1, z = 2k + 3, k \in \mathbb{R}$ and $\dfrac{x - 1}{2} = \dfrac{y - 2}{1} = \dfrac{z - 3}{1}$ is

The three points $A(2, 4, 3), B(4, a, 9)$ and $C(10, -1, 7)$ form a right-angled triangle with $\angle B = 90^\circ$, then the value of "a" is

The distance of the point $\mathrm{P}(-3,4,5)$ from yz plane is

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

If a line makes angles $90^{\circ}, 60^{\circ}$ and $\theta$ with $\mathrm{x}, \mathrm{y}$ and z axes respectively, where $\theta$ is acute, then the value of $\theta$ is

The plane containing the point $(3,2,0)$ and the line $\frac{x-3}{1}=\frac{y-6}{5}=\frac{z-4}{4}$ is

The equation $x y=0$ in three-dimensional space represents

If lines $\frac{x-1}{-3}=\frac{y-2}{2 k}=\frac{z-3}{2}$ and $\frac{x-1}{3 k}=\frac{y-5}{1}=\frac{z-6}{-5}$ are mutually perpendicular, then $k$ is equal to

The distance between the two planes $2 x+3 y+4 z=4$ and $4 x+6 y+8 z=12$ is

The sine of the angle between the straight line $\frac{x-2}{3}=\frac{y-3}{4}=\frac{4-z}{-5}$ are the plane $2 x-2 y+z=5$ is

If \((2,3,-1)\) is the foot of the perpendicular from \((4,2,1)\) to a plane, then the equation of the plane is

The point of intersection of the line \(x+1=\frac{y+3}{3}=\frac{-z+2}{2}\) with the plane \(3 x+4 y+5 z=10\) is

The equation of the plane through the points \((2,1,0),(3,2,-2)\) and \((3,1,7)\) is

The length of perpendicular drawn from the point \((3,-1,11)\) to the line \(\frac{x}{2}=\frac{y-2}{3}=\frac{z-3}{4}\) is

If a line makes an angle of \(\frac{\pi}{3}\) with each \(X\) and \(Y\) axis, then the acute angle made by \(\mathrm{Z}\)-axis is

The distance of the point whose position vector is \((2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}})\) from the plane \(\mathbf{r} \cdot(\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+4 \hat{\mathbf{k}})=4\) is

The angle between the pair of lines \(\frac{x+3}{3}=\frac{y-1}{5}=\frac{z+3}{4}\) and \(\frac{x+1}{1}=\frac{y-4}{4}=\frac{z-5}{2}\) is

The coordinates of foot of the perpendicular drawn from the origin to the plane \(2 x-3 y+4 z=29\) are

The octant in which the point (2, \(-\)4, \(-7\)) lies is

The area of the quadrilateral \(A B C D\) when \(A(0,4,1), B(2,3,-1), C(4,5,0)\) and \(D(2,6,2)\) is equal to

The mid points of the sides of triangle are \((1,5,-1)(0,4,-2)\) and \((2,3,4)\) then centroid of the triangle

The angle between the lines whose direction cosines are \(\left(\frac{\sqrt{3}}{4}, \frac{1}{4}, \frac{\sqrt{3}}{2}\right)\) and \(\left(\frac{\sqrt{3}}{4}, \frac{1}{4}, \frac{-\sqrt{3}}{2}\right)\)

The equation of the line joining the points \((-3,4,11)\) and \((1,-2,7)\) is

If a plane meets the coordinate axes at \(A, B\) and \(C\) in such a way that the centroid of \(\triangle A B C\) is at the point \((1,2,3)\), then the equation of the plane is

If a line makes an angle of with each of \(X\) and \(Y\)-axis, then the acute angle made by Z-axis is

The sine of the angle between the straight line \(\frac{x-2}{3}=\frac{3-y}{-4}=\frac{z-4}{5}\) and the plane \(2 x-2 y+z=5\) is

The point \((1,-3,4)\) lies in the octant

The distance of the point \((1,2,-4)\) from the line \(\frac{x-3}{2}=\frac{y-3}{3}=\frac{z+5}{6}\) is

The distance of the point \((1,2,1)\) from the line \(\frac{x-1}{2}=\frac{y-2}{1}=\frac{z-3}{2}\) is.

\(X Y\) plane divides the line joining the points \(A(2,3,-5)\) and \(B(-1,-2,-3)\) in the ratio

Acute angel between the line \(\frac{(x-5)}{2}=\frac{y+1}{-1}=\frac{z+4}{1}\) and the plane \(3 x-4 y-z+5=0\) is

Foot of the perpendicular drawn from the point \((1,3,4)\) to the plane \(2 x-y+z+3=0\) is

The locus represented by $x y+y z=0$ is

The value of $k$ such that the line $\frac{x-4}{1}=\frac{y-2}{1}=\frac{z-k}{2}$ lies on the plane $2 x-4 y+z=7$ is

The angle between the lines $2 x=3 y=-z$ and $6 x=-y=-4 z$ is

The image of the point $(1,6,3)$ in the line $\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}$ is

The distance of the point $(-2,4,-5)$ from the line $\frac{x+3}{3}=\frac{y-4}{5}=\frac{z+8}{6}$ is

The perpendicular distance of the point $P(6,7,8)$ from $X Y$-plane is

The plane $2 x-3 y+6 z-11=0$ makes an angle $\sin ^{-1}(\alpha)$ with $X$-axis, the value of $\alpha$ is equal to

Reflexion of the point $(\alpha, \beta, \gamma)$ in $X Y$-plane is