Do the vectors (2,0,1),(-2,0,0) and (2,3,0) lie on the same plane? Explain your reasoning.

Expert Answers
justaguide eNotes educator| Certified Educator

Three vectors A, B and C lie on the same plane if one vector can be written as a linear combination of the other two.

To determine if (2,0,1),(-2,0,0) and (2,3,0) lie on the same plane it should be possible to write C1*(2, 0, 1) + C2*(-2, 0, 0) + C3*(2, 3, 0) = 0 with all three of C1, C2 and C3 not being equal to 0.

C1*(2, 0, 1) + C2*(-2, 0, 0) + C3*(2, 3, 0) = 0

2*C1 - 2*C2 + 2*C3 = 0

0*C1 + 0*C2 + 3*C3 = 0

1*C1 + 0*C2 + 0*C3 = 0

It is seen that C1 = 0 and C3 = 0 and C2 = 2.

Therefore the three vectors do not lie not the same plane.