Volume Formula of a Parallelepiped
Formula
The volume of a parallelepiped $V$ can be determined as follows:
[1]: When the area of the base $A$ and the height $h$ are known, it is given by: $$ V = A \times h $$
[2]: When you know the three vectors $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$ that form the parallelepiped in a coordinate space, it is given by: $$ V = |\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})| $$
Alternatively, it is equal to the determinant of a matrix whose column/row vectors are those three vectors. $$ V = \begin{vmatrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{vmatrix} = \begin{vmatrix} a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3} \end{vmatrix} $$
[3]: When you know the magnitudes of the three vectors $a = |\mathbf{a}|$, $b = |\mathbf{b}|$, $c = |\mathbf{c}|$, and the angles between the three vectors $\alpha = \angle(\mathbf{b}, \mathbf{c})$, $\beta = \angle(\mathbf{a}, \mathbf{c})$, $\gamma = \angle(\mathbf{a}, \mathbf{b})$, it is given by: $$ V = abc \sqrt{1 + 2\cos\alpha \cos\beta \cos\gamma - \cos^{2}\alpha - \cos^{2}\beta - \cos^{2}\gamma} $$
Explanation
The formula in [2] is also known as the scalar triple product.
Proof
[2]
Refer to the document on scalar triple product.
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[3]
By [2], the square of the volume is as follows. As the product of determinants is equal to the determinant of a matrix product:
$$ \begin{align*} V^{2} &= \left| \begin{bmatrix} -\ \mathbf{a}\ - \\ -\ \mathbf{b}\ - \\ -\ \mathbf{c}\ - \end{bmatrix} \right| \times \left| \begin{bmatrix} \vert & \vert & \vert \\ \mathbf{a} & \mathbf{b} & \mathbf{c} \\ \vert & \vert & \vert \end{bmatrix} \right| \\[2em] &= \left| \begin{bmatrix} -\ \mathbf{a}\ - \\ -\ \mathbf{b}\ - \\ -\ \mathbf{c}\ - \end{bmatrix} \cdot \begin{bmatrix} \vert & \vert & \vert \\ \mathbf{a} & \mathbf{b} & \mathbf{c} \\ \vert & \vert & \vert \end{bmatrix} \right| \\[2em] &= \left| \begin{bmatrix} \mathbf{a} \cdot \mathbf{a} & \mathbf{a} \cdot \mathbf{b} & \mathbf{a} \cdot \mathbf{c} \\ \mathbf{b} \cdot \mathbf{a} & \mathbf{b} \cdot \mathbf{b} & \mathbf{b} \cdot \mathbf{c} \\ \mathbf{c} \cdot \mathbf{a} & \mathbf{c} \cdot \mathbf{b} & \mathbf{c} \cdot \mathbf{c} \end{bmatrix} \right| \\[2em] &= \left| \begin{bmatrix} a^{2} & ab \cos \gamma & ac \cos \beta \\ ab \cos \gamma & b^{2} & bc \cos \alpha \\ ac \cos \beta & bc \cos \alpha & c^{2} \end{bmatrix} \right| \\[2em] &= a^{2}b^{2}c^{2} \left( 1 + 2\cos\alpha \cos\beta \cos\gamma - \cos^{2}\alpha - \cos^{2}\beta - \cos^{2}\gamma \right) \end{align*} $$
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