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Q21E

Expert-verified
Found in: Page 565

### Essential Calculus: Early Transcendentals

Book edition 2nd
Author(s) James Stewart
Pages 830 pages
ISBN 9781133112280

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# Show the equation $$0 \times {\rm{a}} = 0 = {\rm{a}} \times 0$$ for any vector $${\rm{a}}$$ in $${V_3}$$.

The equation $$0 \times {\rm{a}} = 0 = {\rm{a}} \times 0$$ is shown for any vector $${\rm{a}}$$ in $${V_3}$$.

See the step by step solution

## Step 1: Formula used

Consider, $${\rm{a}} = \left\langle {{a_1},{a_2},{a_3}} \right\rangle$$.

## Step 2: Find the cross product between $$0$$ and $${\rm{a}}$$

$$\begin{array}{l}0 \times {\rm{a}} = \left| {\begin{array}{*{20}{c}}{\rm{i}}&{\rm{j}}&{\rm{k}}\\0&0&0\\{{a_1}}&{{a_2}}&{{a_3}}\end{array}} \right|\\0 \times {\rm{a}} = \left| {\begin{array}{*{20}{c}}0&0\\{{a_2}}&{{a_3}}\end{array}} \right|{\rm{i}} - \left| {\begin{array}{*{20}{c}}0&0\\{{a_1}}&{{a_3}}\end{array}} \right|{\rm{j}} + \left| {\begin{array}{*{20}{c}}0&0\\{{a_1}}&{{a_2}}\end{array}} \right|{\rm{k}}\\0 \times {\rm{a}} = (0 - 0){\rm{i}} - (0 - 0){\rm{j}} + (0 - 0){\rm{k}}\\0 \times {\rm{a}} = 0\end{array}$$

## Step 3: Find the cross product between $${\rm{a}}$$ and $$0$$

Find the cross product between a and 0

$$\begin{array}{l}a \times 0 = \left| {\begin{array}{*{20}{c}}{\rm{i}}&{\rm{j}}&{\rm{k}}\\{{a_1}}&{{a_2}}&{{a_3}}\\0&0&0\end{array}} \right|\\a \times 0 = \left| {\begin{array}{*{20}{c}}{{a_2}}&{{a_3}}\\0&0\end{array}} \right|{\rm{i}} - \left| {\begin{array}{*{20}{c}}{{a_1}}&{{a_3}}\\0&0\end{array}} \right|{\rm{j}} + \left| {\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}\\0&0\end{array}} \right|{\rm{k}}\\a \times 0 = (0 - 0){\rm{i}} - (0 - 0){\rm{j}} + (0 - 0){\rm{k}}\\a \times 0 = 0\end{array}$$

Here, it concludes that$$0 \times {\rm{a}} = 0 = {\rm{a}} \times 0$$.

Therefore, the equation $$0 \times {\rm{a}} = 0 = {\rm{a}} \times 0$$ is shown for any vector $$a$$ in $${V_3}$$.

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