Question

How to do you find the volume of the parallelepiped with the adjacent edges $pq, pr\,and\,ps$ where \[p\left( {3,0,1} \right),{\text{ }}q\left( { - 1,2,5} \right),{\text{ }}r\left( {5,1, - 1} \right){\text{ }}and{\text{ }}s\left( {0,4,2} \right)\].

Answer 1

Hint: Parallelogram means an object having a parallel plane if the adjacent side of the parallelogram are given in the vectors as x, y and z. The volume of a parallelepiped determined by the vectors a, b, c (where a, b and c share the same initial point) is the magnitude of their scalar triple product. Then the volume of such a parallelogram can be calculated by taking the dot product of side x with the cross product of y and z.

Complete step by step answer:
Given data is as below: the coordinates of the sides of the parallelogram are given as –
\[p\left( {3,0,1} \right),{\text{ }}q\left( { - 1,2,5} \right),{\text{ }}r\left( {5,1, - 1} \right){\text{ }}and{\text{ }}s\left( {0,4,2} \right)\]
Now, we will find the three vectors pq, pr and ps using coordinates.So,
\[\overrightarrow {pq} \] = q-p
\[\Rightarrow \overrightarrow {pq}= \left( { - 1,2,5} \right) - \left( {3,0,1} \right)\]
\[\Rightarrow \overrightarrow {pq} = ( - 1 - 3,2 - 0,5 - 1)\]
Simplify the values, we get,
\[\overrightarrow {pq} \]\[ = ( - 4,2,4)\]
Next,
\[\overrightarrow {pr} \]= r - p
\[\Rightarrow \overrightarrow {pr}= \left( {5,1, - 1} \right) - \left( {3,0,1} \right)\]
\[\Rightarrow \overrightarrow {pr}= (5 - 3,1 - 0, - 1 - 1)\]
Simplify the values, we get,
\[ = (2,1, - 2)\]
And,
\[\overrightarrow {ps} \]= s – p
\[\Rightarrow \overrightarrow {ps}= \left( {0,4,2} \right) - \left( {3,0,1} \right)\]
\[\Rightarrow \overrightarrow {ps}= (0 - 3,4 - 0,2 - 1)\]
Simplify the values, we get,
\[ \overrightarrow {ps}= ( - 3,4,1)\]

The scalar triple product is given by the determinant of the matrix \[(3 \times 3)\] that has in the rows the three components of the three vectors:
\[\left| { - 4 + 2 + 4} \right|\]
\[\Rightarrow \left| { + 2 + 1 - 2} \right|\]
\[\Rightarrow \left| { - 3 + 4 + 1} \right|\]
The expression to calculate the volume is given
\[V = \left| {PS \cdot \left( {PQ \times PR} \right)} \right|\]
Therefore, these three vectors can be used to calculate the volume of parallelogram as the triple product that can be expressed in determinant as follow,
\[V = |a \cdot (b \times c)|\]
\[\Rightarrow V = \left| {\begin{array}{*{20}{c}}
  {{a_1}}&{{a_2}}&{{a_3}} \\
  {{b_1}}&{{b_2}}&{{b_3}} \\
  {{c_1}}&{{c_2}}&{{c_3}}
\end{array}} \right| \\
\Rightarrow V= {a_1}[{b_2}{c_3} - {c_2}{b_3}] - {a_2}[{b_1}{c_3} - {b_3}{c_1}] + {a_3}[{b_1}{c_2} - {b_2}{c_1}] \\
 \]
Substituting the values in the determinant form as,
\[D = \left| {\begin{array}{*{20}{c}}
  { - 4}&2&4 \\
  2&1&{ - 2} \\
  { - 3}&4&1
\end{array}} \right|\]
The volume can be calculated in the determinant form as,
\[V = - 4[1(1) - 4( - 2)] - 2[2(1) - ( - 2)( - 3)] + 4[2(4) - 1( - 3)]\]
Simplify the values, we get,
\[V = - 4[1 + 8] - 2[2 - 6] + 4[8 + 3]\]
\[\Rightarrow V = - 4(9) - 2( - 4) + 4(11)\]
\[\Rightarrow V= - 36 + 8 + 44 \\
\Rightarrow V= 16 \\ \]
\[\therefore \left| V \right| = \left| {16} \right| = 16\]

Thus, the volume of the parallelepiped is given as \[16\].

Note: Parallelepiped is a 3-D shape whose faces are all parallelograms. The volume of a parallelepiped is equal to the product of its surface area and height. The dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers, and returns a single number.