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In linear algebra, it is known that the scalar product $$ V = | (\vec{a} \times \vec{b}) \cdot \vec{c} | = \left| \begin{array}{ccc} c_1 & c_2 & c_3 \\[1mm] a_1 & a_2 & a_3 \\[1mm] b_1 & b_2 & b_3 \\[1mm] \end{array} \right| $$ represents the volume of the parallelopiped with $\vec{a}$, $\vec{b}$ and $\vec{c}$ as the three sides.

  1. How to prove this result geometrically?

  2. If $V = 0$, then the volume $V$ of the parallelopiped is zero. This means that $\vec{a}$, $\vec{b}$ and $\vec{c}$ are coplanar.

Vector Algebra

Dr. Sundar
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1 Answers1

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In general, $$V=\text{Area}\cdot \text{height}=\underbrace{|\vec{a}\times \vec{b}|}_{\text{Area}}\cdot \underbrace{|\vec{c}|\cdot \cos(\phi)}_{\text{height}}=(\vec{a}\times \vec{b})\cdot \vec{c}$$

enter image description here

math_undergrad_questions
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  • Good, but there is a mistake in the picture: it seems to suggest that the face the contains $\mathbf a$ and $\mathbf c$ is vertical, while it's not true in general, and the angle $\phi$ is not measured in a plane that contains $\mathbf a$ (i.e., $\mathbf a$, $\mathbf c$ and $\mathbf a\times \mathbf b$ are not coplanar). – Jean-Claude Arbaut Apr 02 '22 at 12:52
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    You're right. See here for a better picture: https://www.mathelike.de/media/k2/items/cache/37d163b88a4522bd852de06260df3d98_L.jpg – math_undergrad_questions Apr 02 '22 at 13:19
  • Hi @racine_carree, this is unrelated to your answer, hence I would be deleting this comment soonest. Please could you help out with this question https://math.stackexchange.com/q/4435979/585488 – linker Apr 29 '22 at 19:15