If a = 2i + 2j + 3k , b = -i + 2j + k and c = 3i + j are such that a + γ b is perpendicular to c , then determine the value of γ ? - Mathematics MCQ Mathematics Mock Test - 9 2026 | ExamTest.live

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If a = 2 \hat{i} + 2 \hat{j} + 3 \hat{k}, b = - \hat{i} + 2 \hat{j} + \hat{k} and c = 3 \hat{i} + \hat{j} are such that a + γ b is perpendicular to c, then determine the value of γ ?

Solution & Step-by-step Explanation

First, find the vector a + γ b : a + γ b = (2 \hat{i} + 2 \hat{j} + 3 \hat{k}) + γ (- \hat{i} + 2 \hat{j} + \hat{k}) = (2 - γ) \hat{i} + (2 + 2 γ) \hat{j} + (3 + γ) \hat{k} Two vectors are perpendicular if their dot product is zero. (a + γ b) \cdot c = 0 [(2 - γ) \hat{i} + (2 + 2 γ) \hat{j} + (3 + γ) \hat{k}] \cdot (3 \hat{i} + \hat{j} + 0 \hat{k}) = 0 3 (2 - γ) + 1 (2 + 2 γ) + 0 (3 + γ) = 0 6 - 3 γ + 2 + 2 γ = 0 8 - γ = 0 ⟹ γ = 8

Try it yourself before checking the explanation above.

If a = 2 \hat{i} + 2 \hat{j} + 3 \hat{k}, b = - \hat{i} + 2 \hat{j} + \hat{k} and c = 3 \hat{i} + \hat{j} are such that a + γ b is perpendicular to c, then determine the value of γ ?

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