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

Hint: 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,
\[\begin{align}
  & v=\overrightarrow{A.}(\overrightarrow{B}\times \overrightarrow{C)}=\det \left( \begin{matrix}
   1 & 1 & 1 \\
   2 & 4 & -1 \\
   1 & 1 & 3 \\
\end{matrix} \right)=(1\times 13)-(1\times 7)+(1\times -2)=4 \\
 & s=|\overrightarrow{A}\times \overrightarrow{B|}=\det \left( \begin{matrix}
   {\hat{i}} & {\hat{j}} & {\hat{k}} \\
   1 & 1 & 1 \\
   2 & 4 & -1 \\
\end{matrix} \right)=|-5\hat{i}+3\hat{j}+2\hat{k}|=\sqrt{{{5}^{2}}+{{3}^{2}}+{{2}^{2}}}=\sqrt{38} \\
 & h=\dfrac{v}{s}=\dfrac{4}{\sqrt{3}8}=\dfrac{2\sqrt{38}}{19} \\
\end{align}\]

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.