# Geometric interpretation of the scalar triple product

The Wikipedia article for scalar triple product says the following:

Geometrically, the scalar triple product $$\mathbf{a}\cdot(\mathbf{b}\times \mathbf{c})$$ is the (signed) volume of the parallelepiped defined by the three vectors given. Here, the parentheses may be omitted without causing ambiguity, since the dot product cannot be evaluated first. If it were, it would leave the cross product of a scalar and a vector, which is not defined.

$$(\mathbf{b}\times \mathbf{c})$$ is the vector normal to vectors $$\mathbf{b}$$ and $$\mathbf{c}$$. What is the geometric reasoning that leads us to understand that the dot product of $$\mathbf{a}$$ with this normal vector is equal to the volume of the parallelepiped defined by the three vectors? I would greatly appreciate it if people would please take the time to explain this.

• Does this answer your question? Scalar triple product - why equivalent to determinant? – Raskolnikov Jun 19 '20 at 5:38
• @Raskolnikov But this doesn't give any geometric explanation. – The Pointer Jun 19 '20 at 5:41
• The length of $b\times c$ is the area of the parallelogram spanned by $b$ and $c$. The scalar product with $a$ projects $a$ onto the line along $b\times c$ which is orthogonal to the parallelogram and, hence, gives the height of the parallelepiped. So, the triple product is nothing but $base \times height$ (up to a sign). – trancelocation Jun 19 '20 at 5:59
• @ThePointer : The dot-product does incorporate the magnitude of both vectors involved: $a \cdot b = |a||b|\cos \angle (a,b)$. – trancelocation Jun 19 '20 at 6:10