Find the distance between the points $P\left(1, -3, 4\right)$ and $Q\left(-4, 1, 2\right)$.

Important Questions on Introduction to Three Dimensional Geometry

MEDIUM

Mathematics>Coordinate Geometry>Introduction to Three Dimensional Geometry>Distance Formula in 3D

If the vertices of the triangles are $(1,2,3),(2,3,1),(3,1,2)$ and if $H, G, S$ and $I$ respectively denote its orthocenter, centroid, circumcenter and incenter, then $H+G+S+I=$

MEDIUM

Mathematics>Coordinate Geometry>Introduction to Three Dimensional Geometry>Distance Formula in 3D

If the vectors $\overrightarrow{AB}=-3\hat{i}+4\hat{k}$ and $\overrightarrow{AC}=5\hat{i}-2\hat{j}+4\hat{k}$ are the sides of a triangle $ABC$. Then the length of the median through $A$ is

MEDIUM

Mathematics>Coordinate Geometry>Introduction to Three Dimensional Geometry>Distance Formula in 3D

The position vectors of the points $A$ and $B$ with respect to $O$ are $2\hat{i}+2\hat{j}+\hat{k}$ and $2\hat{i}+4\hat{j}+4\hat{k}$. The length of the internal bisector of $\angle BOA$ of $∆AOB$ is (take proportionality constant is $2$)

MEDIUM

Mathematics>Coordinate Geometry>Introduction to Three Dimensional Geometry>Distance Formula in 3D

Find the coordinates of the points which divide the line segment joining the points $\left(2, -4, 6\right)$ and $\left(6, 8, -10\right)$ in the ratio $3:4$ internally and externally.

MEDIUM

Mathematics>Coordinate Geometry>Introduction to Three Dimensional Geometry>Distance Formula in 3D

If a point $R\left(4,y,z\right)$ lies on the line segment joining the points $P\left(2,-\mathrm{3,4}\right)$ and $Q\left(\mathrm{8,0},10\right),$ then the distance of $R$ from the origin is

HARD

Mathematics>Coordinate Geometry>Introduction to Three Dimensional Geometry>Distance Formula in 3D

Using division formula prove that points $\left(-4, 6, 10\right), \left(2, 4, 6\right)$ and $\left(14, 0, -2\right)$ are collinear.

MEDIUM

Mathematics>Coordinate Geometry>Introduction to Three Dimensional Geometry>Distance Formula in 3D

What is the ratio in which the line segment joining the points $(-3,2,5)$ and $(6,3,2)$ is divided by the $XY$ - plane?

EASY

Mathematics>Coordinate Geometry>Introduction to Three Dimensional Geometry>Distance Formula in 3D

If the point $\left(a, 8,-2\right)$ divides the line segment joining the points $\left(1,4,6\right)$ and $\left(5,2,10\right)$ in the ratio $m:n$ then $\frac{2m}{n}-\frac{a}{3}=$

EASY

Mathematics>Coordinate Geometry>Introduction to Three Dimensional Geometry>Distance Formula in 3D

$XY-$plane divides the line joining the points $A(2, 3, –5)$ and $B(–1, –2, –3)$ in the ratio

EASY

Mathematics>Coordinate Geometry>Introduction to Three Dimensional Geometry>Distance Formula in 3D

In a triangle $ABC,$ if the mid points of sides $AB,BC,CA$ are $(3,0,0),(0,4,0),(0,0,5)$, respectively, then $A{B}^2+B{C}^2+C{A}^2=$

EASY

Mathematics>Coordinate Geometry>Introduction to Three Dimensional Geometry>Distance Formula in 3D

The shortest distance of the point $(a, b, c)$ from the $x$-axis is:

MEDIUM

Mathematics>Coordinate Geometry>Introduction to Three Dimensional Geometry>Distance Formula in 3D

The plane which bisects the line segment joining the points $\left(-3, -3, 4\right)$ and $\left(3, 7, 6\right)$ at right angles, passes through which one of the following points?

EASY

Mathematics>Coordinate Geometry>Introduction to Three Dimensional Geometry>Distance Formula in 3D

The distance of point $P\left(a, b, c\right)$ from the $x$-axis is

MEDIUM

Mathematics>Coordinate Geometry>Introduction to Three Dimensional Geometry>Distance Formula in 3D

The position vectors of the points $P$ and $Q$ are respectively $-2\overline{i}-3\overline{j}+\overline{k}$ and $3\overline{i}+3\overline{j}+2\overline{k}.$ The ratio in which the point having position vector $\frac{-9}{2}\overline{i}-6\overline{j}+\frac{1}{2}\overline{k}$ divides the line segment joining $P$ and $Q$ is

EASY

Mathematics>Coordinate Geometry>Introduction to Three Dimensional Geometry>Distance Formula in 3D

Let the position vectors of two points $P$ and $Q$ be $2\hat{i}+\hat{j}$ and $\hat{i}-3\hat{j}$ respectively. Then the position vector of a point $R$ which divides the line joining the points $P$ and $Q$ in the ratio $1:2$ externally, is

MEDIUM

Mathematics>Coordinate Geometry>Introduction to Three Dimensional Geometry>Distance Formula in 3D

If the points $A\left(1,3,5\right),B\left(2,4,6\right),C\left(4,5,k\right)$ form a right angled triangle then the number of possible values of $k$ is

EASY

Mathematics>Coordinate Geometry>Introduction to Three Dimensional Geometry>Distance Formula in 3D

If $x$-coordinate of a point $P$ on the line joining the points $Q\left(2,2,1\right)$ and $R\left(5,2,-2\right)$ is $4$, then the $y$-coordinate of $P=$

MEDIUM

Mathematics>Coordinate Geometry>Introduction to Three Dimensional Geometry>Distance Formula in 3D

Let $A$ and $B$ be two points with position vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ respectively and let $C$ be a point dividing $AB$ internally and the position vector of $C$ on $AB$ is $\overrightarrow{c}=\lambda \overrightarrow{a}+\mu \overrightarrow{b},$ then

EASY

Mathematics>Coordinate Geometry>Introduction to Three Dimensional Geometry>Distance Formula in 3D

The point in the $xy-$ plane which is equidistant from $\left(2,0,3\right),\left(0,3,2\right)$ and $\left(0,0,1\right)$ is

EASY

Mathematics>Coordinate Geometry>Introduction to Three Dimensional Geometry>Distance Formula in 3D

The length of the perpendicular drawn from the point $(4,-7,3)$ on the $y-$axis is