# Understanding how to prove that vectors lie on the same plane

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.

• $2b=a+c{}{}{}{}$. – Angina Seng Feb 15 '18 at 7:25

• 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.
• The key point is that $n= b \times c$ is a normal vector to the plane b-c thus if $a\cdot n=0$ also $a$ is in the same plane. – user Feb 15 '18 at 7:58