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.
Asked
Active
Viewed 108 times
0
-
- Which information is hided in the adjective "circumferencial" ? 2) What are your attempts ? Where are you blocked ?
– Jean Marie Jun 14 '19 at 18:45 -
1circumferential part was negligible I just mean that the angles locked between (b,c)and (a,c) are equal. So when I tried to solve it I made a perpendicular from the other end of edge c to the base, defined by a and b,which served as a hight. I got that h= csin(α) and the area of the base - sin(β)ab . I tried to check if it is correct and I made a simulation with some angles and side lengths but I got very different results compared with the product derived from the standard formula for the volume of a parallelepiped V = abc √ (1 + 2cos(α)cos(β)*cos(γ) - cos²(α) - cos²(β) - cos²(γ)). – Hristou Jun 14 '19 at 19:39
1 Answers
1
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 $\det(M^TM)=\det(M)^2$) :
$$V^2=a^2b^2c^2(1+2 \cos^2 \alpha \cos \beta-2\cos^2 \alpha-\cos^2 \beta).$$
Remark : Condition $\alpha > 2\beta$ hasn't been used.
Jean Marie
- 81,803