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Question
If the position vectors of P and Q are \[\hat{i} + 3 \hat{j} - 7 \hat{k} \text{ and } 5 \text{i} - 2 \hat{j} + 4 \hat{k}\] then the cosine of the angle between \[\vec{PQ}\] and y-axis is
Options
\[\frac{5}{\sqrt{162}}\]
\[\frac{4}{\sqrt{162}}\]
\[- \frac{5}{\sqrt{162}}\]
\[\frac{11}{\sqrt{162}}\]
Solution
\[- \frac{5}{\sqrt{162}}\]
\[\vec{PQ} = \vec{OQ} - \vec{OP} = 5 \hat{i} - 2 \hat{j} + 4 \hat{k} - \left( \hat{i} + 3 \hat{j} - 7 \hat{k} \right) = 4 \hat{i} - 5 \hat{j} + 11 \hat{k} \]
\[\text{ The unit vector alongy-axis is } \hat{j} .\]
\[\text{ Let } \theta \text{ be the required angle } . \]
\[\cos \theta = \frac{\vec{PQ} . \hat{j}}{\left| \vec{PQ} \right|\left| \hat{j} \right|} = \frac{- 5}{\sqrt{16 + 25 + 121}\sqrt{1}} = \frac{- 5}{\sqrt{162}}\]
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