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# If the Vertices A, B and C of ∆Abc Have Position Vectors (1, 2, 3), (−1, 0, 0) and (0, 1, 2), Respectively, What is the Magnitude of ∠Abc? - CBSE (Arts) Class 12 - Mathematics

ConceptBasic Concepts of Vector Algebra

#### Question

If the vertices Aand C of ∆ABC have position vectors (1, 2, 3), (−1, 0, 0) and (0, 1, 2), respectively, what is the magnitude of ∠ABC

#### Solution

$\text{ Given that }$
$\vec{OA} = \hat{i} + 2 \hat{j} + 3 \hat{k} ; \vec{OB} = - 1 \hat{i} + 0 \hat{j} + 0 \hat{k} ; \vec{OC} = 0 \hat{i} + 1 \hat{j} + 2 \hat{k}$
$\vec{AB} = \vec{OB} - \vec{OA} = - 2 \hat{i} - 2 \hat{j} - 3 \hat{k} \Rightarrow \left| \vec{AB} \right| = \sqrt{4 + 4 + 9} = \sqrt{17}$
$\vec{BC} = \vec{OC} - \vec{OB} = \hat{i} + \hat{j} + 2 \hat{k} \Rightarrow \left| \vec{BC} \right| = \sqrt{1 + 1 + 4} = \sqrt{6}$
$\vec{CA} = \vec{OA} - \vec{OC} = \hat{i} + \hat{j} + \hat{k} \Rightarrow \left| \vec{CA} \right| = \sqrt{1 + 1 + 1} = \sqrt{3}$
$\cos ∠ ABC = \frac{\left| \vec{AB} . \vec{BC} \right|}{\left| \vec{AB} \right|\left| \vec{BC} \right|} = \frac{\left| - 2 - 2 - 6 \right|}{\left( \sqrt{17} \right)\left( \sqrt{6} \right)} = \frac{10}{\sqrt{102}}$
$\Rightarrow ∠ ABC = \cos^{- 1} \left( \frac{10}{\sqrt{102}} \right)$

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Solution If the Vertices A, B and C of ∆Abc Have Position Vectors (1, 2, 3), (−1, 0, 0) and (0, 1, 2), Respectively, What is the Magnitude of ∠Abc? Concept: Basic Concepts of Vector Algebra.
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