Question

The altitude of a parallelepiped whose three coterminous edges are the vectors, $\overrightarrow{A}=\hat{i}+\hat{j}+\hat{k}$, $\overrightarrow{B}=2\hat{i}+4\hat{j}-\hat{k}$ and $\overrightarrow{C}=\hat{i}+\hat{j}+3\hat{k}$ with $\overrightarrow{A}$ and $\overrightarrow{B}$ as the sides of the base of the parallelepiped is
A. $\dfrac{2}{\sqrt{19}}$
B. $\dfrac{4}{\sqrt{19}}$
C. $\dfrac{2\sqrt{38}}{19}$
D. None

Answer 1

Find the volume of the parallelepiped by taking a scalar triple product of the coterminous vectors. Divide this volume by the area of the base to obtain the altitude.

Complete step by step solution:
The altitude is given by

$altitude(h)=\dfrac{volume(v)}{base\;area(s)}$

Now,

& v=\overrightarrow{A.}(\overrightarrow{B}\times \overrightarrow{C)}=\det \left( \begin{matrix} 1 & 1 & 1 \\\ 2 & 4 & -1 \\\ 1 & 1 & 3 \\\ \end{matrix} \right)=(1\times 13)-(1\times 7)+(1\times -2)=4 \\\ & s=|\overrightarrow{A}\times \overrightarrow{B|}=\det \left( \begin{matrix} {\hat{i}} & {\hat{j}} & {\hat{k}} \\\ 1 & 1 & 1 \\\ 2 & 4 & -1 \\\ \end{matrix} \right)=|-5\hat{i}+3\hat{j}+2\hat{k}|=\sqrt{{{5}^{2}}+{{3}^{2}}+{{2}^{2}}}=\sqrt{38} \\\ & h=\dfrac{v}{s}=\dfrac{4}{\sqrt{3}8}=\dfrac{2\sqrt{38}}{19} \\\ \end{aligned}$$ **The correct answer is option (C)** **Additional information:** A parallelepiped is a three dimensional figure formed by six parallelogram faces. It has 3 sets of 4 equal parallel edges. It has 8 vertices. It is also a zonohedron as each face has point symmetry. They can be obtained by linear transformation of a cube. Cuboid, cube and rhombohedron are specific cases of parallelepiped. A perfect parallelepiped has integer edges, faces diagonals and body diagonals and is known to exist in nature. A parallelotope is a generalization of a parallelepiped in higher dimensions. The volume of any tetrahedron that shares the three coterminous edges of a parallelepiped is one sixth the volume of the parallelepiped. **Note:** The scalar triple product is cyclic in nature. If it is equal to zero then it implies that the three vectors are coplanar. The cross product is anti-commutative and distributive over addition.