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Question
Show that the points whose position vectors are \[\vec{a} = 4 \hat{i} - 3 \hat{j} + \hat{k} , \vec{b} = 2 \hat{i} - 4 \hat{j} + 5 \hat{k} , \vec{c} = \hat{i} - \hat{j}\] form a right triangle.
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Solution
Given that
\[ \vec{a} = \vec{OA} = 4 \hat{i} - 3 \hat{j} + \hat{k} ; \vec{b} = \vec{OB} = 2 \hat{i} - 4 \hat{j} + 5 \hat{k} ; \vec{c} = \vec{OC} = \hat{i} - \hat{j} + 0 \hat{k} \]
\[ \vec{AB} = \vec{OB} - \vec{OA} = - 2 \hat{i} - \hat{j} + 4 \hat{k} \]
\[ \vec{BC} = \vec{OC} - \vec{OB} = - \hat{i} + 3\hat{j} - 5 \hat{k} \]
\[ \vec{CA} = \vec{OA} - \vec{OC} = 3\hat{i} - 2 \hat{j} + \hat{k}] \]
\[ \vec{AB} . \vec{BC} = 2 - 3 - 20 = - 21 \neq 0\]
\[ \vec{BC} . \vec{CA} = - 3 - 6 - 5 = - 14 \neq 0\]
\[ \vec{AB} . \vec{CA} = - 6 + 2 + 4 = 0\]
\[\text{So}, \vec{AB} \text{ is perpendicular to } \vec{CA} .\]
\[\text{So}, ∆ABC\hspace{0.167em}\text{ is a right-angled triangle. }\]
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