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Question
Find \[\vec{a} \cdot \vec{b}\] when
\[\vec{a} = \hat{j} - \hat{k} \text{ and } \vec{b} = 2 \hat{i} + 3 \hat{j} - 2 \hat{k}\]
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Solution
\[ \vec{a} = \hat{j} - \hat{k} = 0 \hat{i} + \hat{j} - \hat{k}\text{ and } \vec{b} =2 \hat{i} + \hat{3j} -2 \hat{k}\]
\[ \vec{a} . \vec{b} =\left( 0 \hat{i} + \hat{j} - \hat{k} \right).\left( 2 \hat{i}+ \hat{3j}-2 \hat{k}\right)\]
\[ = \left( 0 \right)\left( 2 \right) + \left( 1 \right)\left( 3 \right) + \left( - 1 \right)\left( - 2 \right)\]
\[ = 3 + 2\]
\[ = 5\]
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