Unit-I: Sets and Functions
Unit-II: Algebra
Unit-III: Coordinate Geometry
Unit-IV: Calculus
Unit-V: Mathematical Reasoning
Unit-VI: Statistics and Probability

Projection of a segment on a line or a plane

If  P' \text { and } Q' are projections of  P\text { and } Q on a given line or plane, then  \overline{P'Q'} is the projection of the segment  \overline{PQ} on the given line or plane.

Distance between Two Points in Space:

The distance between the points  P (x_1,y_1,z_1) &  Q (x_2,y_2,z_2) is given by

PQ= \sqrt{(x_1-x_2)^{2}+(y_1-y_2)^{2}+(z_1-z_2)^{2}}

Note:

\to Distance of  P(x, y, z) from  origin is given by OP=\sqrt{x^2+y^2+z^2} .

\to If  L,M,N are projection of  P(x, y, z) on x,y \text { and } z -axes, then PL=\sqrt{y^2+z^2},\quad PM=\sqrt{z^2+x^2}, \quad PN=\sqrt{x^2+y^2} respectively.

(i) Example:

Given  P (2, 3, 7)  and   Q (5, 7, -5)   be two points in space. Then

 \begin{aligned} PQ&= \sqrt{(x_2-x_1)^{2}+ (y_2-y_1)^{2}+(z_2-z_1)^{2}}\\&= \sqrt{(5-2)^{2}+ (7-3)^{2}+(-5-7)^{2}} \\&= \sqrt{3^{2}+ 4^{2}+12^{2}} \\&= \sqrt{9+16+144} = \sqrt{169} \\& = 13 \text{ unit} \end{aligned}

(ii) Example:

Given  A (3, -4, 12)  be a point in space, Then its distance from origin  O (0,0,0) is

\begin{aligned} OP &=\sqrt{x^{2}+y^{2}+z^{2}}\\& =\sqrt{3^{2}+(-4)^{2}+(12)^{2}} \\& =\sqrt{5^{2}+(12)^{2}} \\&=\sqrt{13^{2}}=13 \text{ unit} \end{aligned}

(iii) Example:

Let  L, M, N  are projections of  P (4, 7, -3) on  x, y  and  z -axes respectively, then

\begin{aligned} PL&= \sqrt{y^{2}+z^{2}} = \sqrt{7^{2}+(-3)^{2}}\\& = \sqrt{49+9}=\sqrt{58} \end{aligned}

\begin{aligned} PM&= \sqrt{z^{2}+x^{2}}=\sqrt{(-3)^{2}+4^{2}} \\&= \sqrt{9+16}=\sqrt{25}=5 \end{aligned}

 \begin{aligned} PN&= \sqrt{x^{2}+y^{2}}=\sqrt{4^{2}+7^{2}} \\&= \sqrt{16+49}=\sqrt{65} \end{aligned}

 \therefore PM=5 \text { units}

PL=\sqrt{58} \text { units}

PN=\sqrt{65} \text { units}

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