Tag: general form of the equation of a plane

Questions Related to general form of the equation of a plane

Consider the plane passing through the points $A(2, 2, 1), B(3, 4, 2)$ and $C(7, 0, 6)$.
Which one of the following points lies on the plane?

maths vectors:planes in three dimensions cartesian equation of plane general form of the equation of a plane lines in space

  1. $(1, 0, 0)$

  2. $(1, 0, 1)$

  3. $(0, 0, 1)$

  4. None of the above

Show answer
Correct Option: A
Explanation:

The plane passing through the points $A(2,2,1),B(3,4,2) and C(7,0,6)$.


We can get two vectors in the plane by subtracting pairs of points in the
plane :
$[ 2, 2, 1 ] - [ 3 ,4, 2 ] = [ -1, -2, -1 ]$

$[ 7 ,0, 6 ] - [ 3, 4 ,2 ] = [ 4, -4 ,4 ]$

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


$ [ -1 ,-2, -1 ] \times  [ 4 ,-4 ,4 ] = [ 1, 0 ,0 ] $

hence the points lies on the plane of $(1 , 0 , 0)$