Class 11 Maths Introduction To 3D Geometry Section Formula

Section Formula

Let the two given points be P(x1, y1, z1) and Q (x2, y2, z2). Let the point R (x, y, z) divide PQ in the given ratio m : n internally.  Coordinates of Point R will be,

If the point R divides PQ externally in the ratio m : n, then its coordinates are obtained by replacing n by – n so that coordinates of point R will be

Special case: Coordinates of the mid-point: In case R is the mid-point of PQ, then m : n = 1 : 1 .The coordinates of point R which divides PQ in ratio k : 1 are obtained by taking k= m/n

Numerical: Find coordinates of point which divides line segment joining points (1, –2, 3) and (3, 4, –5) in the ratio 2 : 3   internally, and   externally.

Solution: Here m=2 & n=3. Coordinates of Point P when it divided m:n internally  is

Px = (mx2 + nx1) / (m+n)  =  (2* 3 + 3 * 1)/(2+3)  = 9/5

Py = (my2 + ny1) / (m+n) =(2* 4 + 3 * (-2))/(2+3) = 2/5

Pz = (mz2 + nz1) / (m+n) = (2* (-5) + 3 * 3)/(2+3) = -1/5

There coordinates of P are (9/5 , 2/5, -1/5)

Similarly in case the line is divided externally in ration m:n, then coordinates of Point P are

Px = (mx2 - nx1) / (m-n)  =  (2* 3 - 3 * 1)/(2-3)  = -3

Py = (my2 - ny1) / (m-n) =(2* 4 - 3 * (-2))/(2-3) = -14

Pz = (mz2 - nz1) / (m-n) = (2* (-5) - 3 * 3)/(2-3) = 19

There coordinates of P are (-3,-14, 19)

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