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arrow_back Let $S=\{\mathbf{i}, \mathbf{j}, \mathbf{k}\}$ be the set of standard unit vectors in $R^{3} .$ Show that each ordered pair of vectors in $S$ is orthogonal.

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Let $S=\{\mathbf{i}, \mathbf{j}, \mathbf{k}\}$ be the set of standard unit vectors in $R^{3} .$ Show that each ordered pair of vectors in $S$ is orthogonal.

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It suffices to show that $$ \mathbf{i} \cdot \mathbf{j}=\mathbf{i} \cdot \mathbf{k}=\mathbf{j} \cdot \mathbf{k}=\mathbf{0} $$ because it will follow automatically from the symmetry property of the dot product that $$ \mathbf{j} \cdot \mathbf{i}=\mathbf{k} \cdot \mathbf{i}=\mathbf{k} \cdot \mathbf{j}=\mathbf{0} $$ Although the orthogonality of the vectors in $S$ is evident geometrically from Figure $3.2 .2$ it is confirmed algebraically by the computations $$ \begin{aligned} &\mathbf{i} \cdot \mathbf{j}=(1,0,0) \cdot(0,1,0)=0 \\ &\mathbf{i} \cdot \mathbf{k}=(1,0,0) \cdot(0,0,1)=0 \\ &\mathbf{j} \cdot \mathbf{k}=(0,1,0) \cdot(0,0,1)=0 \end{aligned} $$
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