The values of a for which the points A, B, C with position vectors 2i - j + k , i - 3j - 5k and ai - 3j + k respectively are the vertices of a right-a - Mathematics MCQ AIEEE 2006 2026 | ExamTest.live

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The values of a for which the points A, B, C with position vectors 2 \hat{i} - \hat{j} + \hat{k}, \hat{i} - 3 \hat{j} - 5 \hat{k} and a \hat{i} - 3 \hat{j} + \hat{k} respectively are the vertices of a right-angled triangle with ∠ C = π /2 are:

Solution & Step-by-step Explanation

Vectors for the sides: C A = A - C = (2 - a) \hat{i} + (- 1 - (- 3)) \hat{j} + (1 - 1) \hat{k} = (2 - a) \hat{i} + 2 \hat{j} CB = B - C = (1 - a) \hat{i} + (- 3 - (- 3)) \hat{j} + (- 5 - 1) \hat{k} = (1 - a) \hat{i} - 6 \hat{k} Since ∠ C = 9 0^{\circ}, C A \cdot CB = 0 : ((2 - a) \hat{i} + 2 \hat{j}) \cdot ((1 - a) \hat{i} - 6 \hat{k}) = 0 (2 - a) (1 - a) + 2 (0) + 0 (- 6) = 0 (2 - a) (1 - a) = 0 ⟹ a = 2, 1 .

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The values of a for which the points A, B, C with position vectors 2 \hat{i} - \hat{j} + \hat{k}, \hat{i} - 3 \hat{j} - 5 \hat{k} and a \hat{i} - 3 \hat{j} + \hat{k} respectively are the vertices of a right-angled triangle with ∠ C = π /2 are:

2 and 1

- 2 and - 1

- 2 and 1

2 and - 1

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