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Distance Between Two Points

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Distance Between Two Points - Lesson Summary

The distance between two points A (x1,y1) and B (x2,y2)
AB = ( x 2 - x 1 ) 2 + ( y 2 - y 1 ) 2  .

Calculating the distance between two points in space:
Let A and B be two points. Let the coordinates of A be (x1,y1,z1) and B be (x2,y2,z2)
The distance between points A and B denoted by AB is to be found.

Consider the planes passing through points A and B, and parallel to the coordinate planes. A rectangular parallelepiped is obtained from the intersecting planes. Take a point C on one vertex of the parallelepiped.

Since points A and C lie on the same line parallel to the Y-axis, point C will have the same x and z coordinates, as point A. Hence, the coordinates of point C are (x1,y2,z1).

The triangle formed by points A, B and C is a right-angled triangle with right angle at C.

Since ACB is a right-angled triangle, we get AB2 = AC2 + BC2.

Next, we consider point D on the parallelepiped.

Since points B and D lie on the same line parallel to the Z-axis, its x and y coordinates are the same as that of point B. Hence, the coordinates of point D are (x2,y2,z1). Points D, B and C form a right-angled triangle with a right angle at D.

Therefore, from ΔBDC, BC2 = DC2 + DB2.

In triangles ACB and BDC, we have side CB as common.

Therefore, the relations can be combined as AB2 = AC2 + DC2 + DB2.

Length of AC is given by y2 - y1.

AB2 = (y2 - y1)2 + DC2 + DB2.

Similarly, length of DC is given by x2 - x1 and DB is given by z2 - z1.

AB2 = (y2 - y1)2 + (x2 - x1)2 + (z2 - z1)2.

Hence, the distance between any two points, AB = ( x 2 - x 1 ) 2 + ( y 2 - y 1 ) 2 + ( z 2 - z 1 ) 2

The distance of a point from the origin:
Origin: (0,0,0), Point:P(x,y,z)
OP = √((x - 0)2 + (y - 0)2 + (z - 0)2) = √(x2 + y2 + z2).


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