The edges of parallelepiped that have a common vertex are a, b and c. The angle between a and b is β, and the angle that the circumferential edge c forms with each of the edges a and b is acute with a magnitude of α with α>2β. Find the volume of the parallelepiped.
Volume of parallelepiped with undefined angles
volume
Best Answer
Let us denote by $\vec{A},\vec{B},\vec{C}$ the vectors issued from the common vertex, with resp. length (=norms) $a,b,c$.
Let us define matrix
$$M=[\vec{A}|\vec{B}|\vec{C}]$$
(in the RHS, columns entries are the coordinates of $\vec{A},\vec{B},\vec{C}$ with respect to a certain orthonormal basis).
It is well known that $\det(M)=V$ (parallelepiped's volume).
Now compute (in close connection with the computations of paragraph "volume" in this reference) the so-called "Gram matrix" of ordered system $(\vec{A},\vec{B},\vec{C})$:
$$M^T M=\begin{pmatrix}\vec{A}\cdot\vec{A}&\vec{A}\cdot\vec{B}&\vec{A}\cdot\vec{C}\\ \vec{B}\cdot\vec{A}&\vec{B}\cdot\vec{B}&\vec{B}\cdot\vec{C}\\ \vec{C}\cdot\vec{A}&\vec{C}\cdot\vec{B}&\vec{C}\cdot\vec{C} \end{pmatrix}$$
where the dots are for dot product.
Using the given angles :
$$M^T M=\begin{pmatrix}a^2&ab \cos \beta&ac \cos \alpha\\ ab \cos \beta&b^2&bc \cos \alpha\\ ac \cos \alpha&bc \cos \alpha&c^2 \end{pmatrix}$$
Equating the determinants of the LHS and the RHS (remember that $\det(M^TM)=\det(M)^2$) :
Remark : Condition $\alpha > 2\beta$ hasn't been used.