In order to locate the position of point in a plane or two dimensions, we require a pair of coordinate axis. The distance of the point from the centre is called x-coordinate (or abscissa) and the distance of the point from is called y-coordinate (or ordinate). The ordered pair (x,y) represents co-ordinate of the point. Consider two points \(A(x_{1},y_{1})\;and\;B(x_{2},y_{2})\)

**\(d=\sqrt{(x_{2}-x_{1})^2+(y_{2}-y_{1})^2}\)**

This study can be extended to determine the distance of two points in space. Let the points \(P(x_{1},y_{1},z_{1})\)

Through the points P and Q, we draw planes parallel to the rectangular co-ordinate plane such that we get a rectangular parallelepiped with PQ as the diagonal. ∠PAQ forms a right angle and therefore, using the Pythagoras theorem in triangle PAQ ,

\(PQ^2= PA^2+AQ^2\)

Also, in triangle ANQ, ∠ANQ is a right angle. Similarly, applying the Pythagoras theorem in ΔANQ we get,

\(AQ^2=AN^2+NQ^2\)

From the equations 1 and 2 we have,

\(PQ^2=PA^2+NQ^2+AN^2\)

As co-ordinates of the points P and Q are known,

\(PA=y_{2}-y_{1}\)

Therefore,

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

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

This formula gives us the distance between two points \(P(x_1,y_1,z_1)\)

Distance of any point \(Q(x,y,z)\)

\(OQ=\sqrt{(x^2+y^2+z^2)}\)

Let us go through some examples to understand the distance formula in three dimensions.

\(PQ=\sqrt{((x_2-x_1 )^2+(y_2-y_1 )^2+(z_2-z_1 )^2 )}\) \(PQ=\sqrt{(6-2)^2+(4-(-8))^2+(-3-3)^2}\) \(PQ=\sqrt{(16+144+81)}\) PQ=15.524
Co-ordinates of the points A,B and C are given as (a,0,0), (0,b,0), (0,0,c) and (0,0,0). As we know that the point P is equidistant from the given points. Hence, PA = PB = PC = PO Now, applying the distance formula for PO = PA, we get \(\sqrt{x^2+y^2+z^2}=\sqrt{(x-a)^2+y^2+z^2}\) \(x^2+y^2+z^2=(x-a)^2+y^2+z^2\) \(x^2=(x-a)^2\) \(x= a/2\) Similarly applying the distance formula for PO = PB and PO=PC, we get \(y= \frac{b}{2}\) Therefore co-ordinates of the point which are equidistant from the points A,B,C and O is given by\((\frac{a}{2},\;\frac{b}{2},\;\frac{c}{2})\) |

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