Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 13 - Vector Geometry - 13.4 The Cross Product - Exercises - Page 678: 42


$a\sqrt {c^{2}+b^{2}}$

Work Step by Step

The area of the parallelogram is given as: $||⟨a,0,0⟩\times ⟨0,b,c⟩||$ Thus we have: $⟨a,0,0⟩\times ⟨0,b,c⟩=\begin{vmatrix}\textbf{i}&\textbf{j}&\textbf{k}\\a&0&0\\0&b&c\end{vmatrix}$ $=\textbf{i}(0-0)-\textbf{j}(ac-0)+\textbf{k}(ab-0)$ $=-ac\textbf{j}+ab\textbf{k}$ $||⟨a,0,0⟩\times ⟨0,b,c⟩||=$ $\sqrt {(-ac)^{2}+(ab)^{2}}=\sqrt {a^{2}c^{2}+a^{2}b^{2}}$ $=\sqrt {a^{2}(c^{2}+b^{2})}=a\sqrt {c^{2}+b^{2}}$
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