Image point of $\left(1,\mathrm{ }\mathrm{3,4}\right)$ in the plane $2x-y+z+3=0$ is

Important Questions on Three Dimensional Geometry

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Line and a Point

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

HARD

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Line and a Point

The foot of the perpendicular drawn from the point $\left(4, 2, 3\right)$ to the line joining the points $\left(1, -2, 3\right)$ and $\left(1, 1, 0\right)$ lies on the plane

HARD

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Line and a Point

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

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Line and a Point

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

HARD

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Line and a Point

The distance of the point having position vector $-\widehat{i}+2\widehat{j}+6\widehat{k}$ from the straight line passing through the point $\left(2,\mathrm{ }3,\mathrm{ }-4\right)$ and parallel to the vector, $6\widehat{i}+3\widehat{j}-4\widehat{k}$ is

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Line and a Point

The length of perpendicular from the point $\hat{\mathrm{i}}+2\hat{\mathrm{j}}+3\hat{\mathrm{k}}$ to the line $\frac{x-6}{3}=\frac{y-7}{2}=\frac{z-7}{-2}$ is

HARD

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Line and a Point

The length of perpendicular drawn from the point $(2,3,4)$ to line $\frac{4-x}{2}=\frac{y}{6}=\frac{1-z}{3}$, is

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Line and a Point

Let the foot of perpendicular from a point $P(1,2,-1)$ to the straight line $L:\frac{x}{1}=\frac{y}{0}=\frac{z}{-1}$ be $N$. Let a line be drawn from $P$ parallel to the plane $x+y+2z=0$ which meets $L$ at point $Q$. If $\alpha$ is the acute angle between the lines $PN$ and $PQ$, then $\cos \alpha$ is equal to              .

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Line and a Point

If the length of the perpendicular from the point $\left(\beta ,0,\beta \right),\left(\beta \ne 0\right)$ to the line, $\frac{x}{1}=\frac{y-1}{0}=\frac{z+1}{-1}$ is $\sqrt{\frac{3}{2}}$ , then $\beta$ is equal to

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Line and a Point

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

EASY

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Line and a Point

Let $a,b\in R.$ If the mirror image of the point $P\left(a,6,9\right)$ with respect to the line $\frac{x-3}{7}=\frac{y-2}{5}=\frac{z-1}{-9}$ is $\left(20, b,-a-9\right),$ then $|a+b|$ is equal to:

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Line and a Point

The length of the perpendicular from the point $(2,-1, 4)$ on the straight line $\frac{x+3 }{10}=\frac{y-2}{-7}=\frac{z}{1}$ is

HARD

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Line and a Point

The shortest distance between $A(1, 0, 2)$ and the line $\frac{x+1}{3}=\frac{y-2}{-2}=\frac{z+1}{-1}$ is given by line joining $A$ and $B$, then $B$ in the line is

HARD

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Line and a Point

If the position vectors of three points $A, B, C$ respectively are $\hat{\mathrm{i}}+2\mathrm{j}+\hat{\mathrm{k}}, 2\hat{\mathrm{i}}-\hat{\mathrm{j}}+2\hat{\mathrm{k}}$ and $\hat{\mathrm{i}}+\hat{\mathrm{j}}+2\hat{\mathrm{k}}$, then the perpendicular distance of the point $C$ from the line $AB$ is

HARD

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Line and a Point

Consider the line $L$ given by the equation $\frac{x-3}{2}=\frac{y-1}{1}=\frac{z-2}{1}.$ Let $Q$ be the mirror image of the point $(2,3,-1)$ with respect to $L.$ Let a plane $P$ be such that it passes through $Q,$ and the line $L$ is perpendicular to $P.$ Then which of the following points is on the plane $P?$

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Line and a Point

What will be the distance of point $(2, 4, -1)$ from the straight line given by $(x+5)=(y+3) / 4=(6-z)/9 ?$

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Line and a Point

The length of the perpendicular from the point $\left(1,2,3\right)$ to the line $\frac{-x+6}{-3}=\frac{y-7}{2}=\frac{z-7}{-2}$ is _________.

HARD

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Line and a Point

If $(a, b, c)$ is the image of the point $(1,2,-3)$ in the line, $\frac{x+1}{2}=\frac{y-3}{-2}=\frac{z}{-1},$ then $a+b+c$ is equal to:

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Line and a Point

If the foot of the perpendicular drawn from the point $\left(1,0,3\right)$ on a line passing through $\left(\alpha ,\mathrm{7,1}\right)$ is $\left(\frac{5}{3},\frac{7}{3},\frac{17}{3}\right)$, then $\alpha$ is equal to

HARD

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Line and a Point

The vertices $B$ and $C$ of a $\Delta ABC$ lie on the line, $\frac{x+2}{3}=\frac{y-1}{0}=\frac{z}{4}$ such that $BC=5$ units. Then the area (in sq. units) of this triangle, given the point $A\left(1,-1,2\right),$ is