Question

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?

• A. $(1,0,0)$
• B. $(1,0,1)$
• C. $(0,0,1)$
• D. None of the above

Answer 1

The correct option is A $(1,0,0)$

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

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 $(1,0,0)$