Understanding how to prove that vectors lie on the same plane

Question

I am trying to show that vectors $a= i + 2j +3k$, $b = 4i + 5j +6k$ , $c= 7i +8j+9k$ all lie in the same plane.

I have looked up online and saw that to show that, I have to show that -

$a\cdot (b \times c) = 0$ , using both the cross and dot product.

I am not sure in the understanding behind why I need to prove that to show that all 3 vectors lie in the same plane.

Answer 1

Yes triple product is a correct metod to verify whether or not three vectors lie in the same plane, indeed

• cross product $\vec b\times \vec c\,$ gives a vector normal to the plane and
• $\vec a\cdot (b\times c)=0\,$ gives the conditon that also $\vec a$ is in the plane of $\vec b$ and $\vec c$

More in general note that the triple product $\vec a\cdot (\vec b\times \vec c)$ is the (signed) volume of the parallelepiped defined by the three vectors given, thus it is equal to zero if and only if the three vectors lie on the same plane.

As an alternative we can also verify whether or not the three vectors are linearly dependent by inspection or by the standard method of matrix RREF.