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.

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

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.

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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.

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  • $\begingroup$ But if it’s equals to 0. the theta of the dot product is 9@ degrees. Meaning it’s perpendicular. How does showing that they are perpendicular mean that they are on the same plane ? Sorry I have difficulty visualising this.. $\endgroup$ – user175089 Feb 15 '18 at 7:55
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    $\begingroup$ 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. $\endgroup$ – user Feb 15 '18 at 7:58

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