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Expert-verified Found in: Page 565 ### Essential Calculus: Early Transcendentals

Book edition 2nd
Author(s) James Stewart
Pages 830 pages
ISBN 9781133112280 # To find: The volume of the parallelepiped determined by the vectors a, b and c.

The volume of the parallelepiped determined by the vectors a, b and $$c$$ is 1 cubic units.

See the step by step solution

## Step 1: Concept of parallelepiped with the help of Formula’s

Consider two three-dimensional vectors such as,

$$\begin{array}{l}a = \left\langle {{a_1},{a_2},{a_3}} \right\rangle \\{\rm{b}} = \left\langle {{b_1},{b_2},{b_3}} \right\rangle \end{array}$$

Write the expression for cross product between $$a$$ and $${\rm{b}}$$ vectors.

$$a \times b = \left| {\begin{array}{*{20}{c}}{\rm{i}}&{\rm{j}}&{\rm{k}}\\{{a_1}}&{{a_2}}&{{a_3}}\\{{b_1}}&{{b_2}}&{{b_3}}\end{array}} \right|(1)$$

Write the expression for dot product between $$a$$ and $${\rm{b}}$$ vectors.

$$a \cdot b = {a_1}{b_1} + {a_2}{b_2} + {a_3}{b_3}(2)$$

Write the expression to calculate the parallelepiped's volume.

$$V = |{\rm{a}} \cdot ({\rm{b}} \times {\rm{c}})|(3)$$

## Step 2: Volume of parallelepiped by calculating through Formula.

$$a = {\rm{i}} + {\rm{j}},{\rm{b}} = {\rm{j}} + {\rm{k and }}c = i + j + k$$

Modify equation ($$1$$).

$$b \times c = \left| {\begin{array}{*{20}{c}}{\rm{i}}&{\rm{j}}&{\rm{k}}\\{{b_1}}&{{b_2}}&{{b_3}}\\{{c_1}}&{{c_2}}&{{c_3}}\end{array}} \right|$$

We can substitute $$0$$ for $${b_1},1$$ for $${b_2},1$$ for $${b_3},1$$ for $${c_1},1$$ for $${c_2}$$ and $$1$$ for $${c_3}$$,

$$\begin{array}{l}b \times c = \left| {\begin{array}{*{20}{c}}{\rm{i}}&{\rm{j}}&{\rm{k}}\\0&1&1\\1&1&1\end{array}} \right|\\b \times c = \left| {\begin{array}{*{20}{l}}1&1\\1&1\end{array}} \right|{\rm{i}} - \left| {\begin{array}{*{20}{l}}0&1\\1&1\end{array}} \right|{\rm{j}} + \left| {\begin{array}{*{20}{l}}0&1\\1&1\end{array}} \right|{\rm{k}}\\b \times c = (1 - 1){\rm{i}} - (0 - 1){\rm{j}} + (0 - 1){\rm{k}}\\b \times c = 0{\rm{i}} + 1{\rm{j}} - 1{\rm{k}}\end{array}$$

Modify equation $$\left( 2 \right)$$

$$a \cdot (b \times c) = {a_1}\left( {{b_1} \times {c_1}} \right) + {a_2}\left( {{b_2} \times {c_2}} \right) + {a_3}\left( {{b_3} \times {c_3}} \right)$$

We are substituting $$1$$ for $${a_1},1$$ for $${a_2},0$$ for $${a_3},0$$ for $$\left( {{b_1} \times {c_1}} \right),1$$ for $$\left( {{b_2} \times {c_2}} \right)$$ and $$- 1$$ for $$\left( {{b_3} \times {c_3}} \right)$$,

$$\begin{array}{l}a \cdot (b \times c) = 1(0) + 1(1) + 0( - 1)\\a \cdot (b \times c) = 0 + 1 + 0\\a \cdot (b \times c) = 1\end{array}$$

We are substituting $$1$$ for $$a \cdot (b \times c)$$ in equation $$(3)$$

$$\begin{array}{l}V = |1|\\V = 1\end{array}$$

Thus, the volume of the parallelepiped determined by the vectors $$a$$, $$b$$ and $$c$$ is $$1$$ cubic units. ### Want to see more solutions like these? 