Find the equation of the plane through the three points 2410

Find the equation of the plane through the three points, (2,4,10), (8,13,16) and (5,7,17)

Solution

We can get two vectors in the plane by subtracting pairs of points in the plane:

(2,4,10), (8,13,16) and (5,7,17) : ( 8, 13, 16) - ( 2 , 4 , 10) and ( -5 , 7 , 17) - (2. 4 , 10)

= ( 6, 9 , 6) and (-7 . 3 , 7 )

The cross product of these two vectors will be in the unique direction orthogonal to both, and hence in the direction of the normal vector to the plane.

( 6, 9 , 6) x (-7 . 3 , 7 ) = ( 45 , -84 , 81)

Equation of plane can be given as : a(x - xo) + b(y- yo) + c( z - zo) =0

45(x - 2) -84 (y- 4) + 10( z - 81) =0

45x -84y +10z -90 +336 -810 =0

45x -84y +10z = 564     equation of plane