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Find the Angle Between the Vectors → a and → B → a = 2 ^ I − ^ J + 2 ^ K and → B = 4 ^ I + 4 ^ J − 2 ^ K - CBSE (Arts) Class 12 - Mathematics

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Question

Find the angle between the vectors \[\vec{a} \text{ and } \vec{b}\]  \[\vec{a} = 2\hat{i} - \hat{j} + 2\hat{k} \text{ and } \vec{b} = 4\hat{i} + 4 \hat{j} - 2\hat{k}\]

Solution

\[\text { Let }\theta\text{ be the angle between } \vec{a} \text{ and } \vec{b} . \]

\[\left| \vec{a} \right| = \sqrt{\left( 2 \right)^2 + \left( - 1 \right)^2 + \left( 2 \right)^2} = \sqrt{9} = 3\]

\[\left| \vec{b} \right| = \sqrt{\left( 4 \right)^2 + \left( 4 \right)^2 + \left( - 2 \right)^2} = \sqrt{36} = 6\]

\[ \vec{a} . \vec{b} = 8 - 4 - 4 = 0\]

\[\cos \theta = \frac{\vec{a} . \vec{b}}{\left| \vec{a} \right| \left| \vec{b} \right|} = \frac{0}{\left( 3 \right)\left( 6 \right)} = 0\]

\[ \Rightarrow \theta = \cos^{- 1} \left( 0 \right) = \frac{\pi}{2}\]

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Solution Find the Angle Between the Vectors → a and → B → a = 2 ^ I − ^ J + 2 ^ K and → B = 4 ^ I + 4 ^ J − 2 ^ K Concept: Basic Concepts of Vector Algebra.
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