# If \vec{a}=\hat{i}-\hat{j}+3\hat{k}, \,\vec{b}=5\hat{i}-2\hat{j}+\hat{k} \,and \,\vec{c}=\hat{i}-\hat{j} are such that \vec{a}+μ\vec{b} is perpendicular to \vec{c}, then the value of μ.

Category: QuestionsIf \vec{a}=\hat{i}-\hat{j}+3\hat{k}, \,\vec{b}=5\hat{i}-2\hat{j}+\hat{k} \,and \,\vec{c}=\hat{i}-\hat{j} are such that \vec{a}+μ\vec{b} is perpendicular to \vec{c}, then the value of μ.
Editor">Editor Staff asked 11 months ago

If \vec{a}=\hat{i}-\hat{j}+3\hat{k}, \,\vec{b}=5\hat{i}-2\hat{j}+\hat{k} \,and \,\vec{c}=\hat{i}-\hat{j} are such that \vec{a}+μ\vec{b} is perpendicular to \vec{c}, then the value of μ.

(a) \frac{7}{2}

(b) –\frac{7}{2}

(c) –\frac{3}{2}

(d) \frac{7}{9}

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This intriguing question originated from Product of Two Vectors-2 topic in portion Vector Algebra of Mathematics – Class 12
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Editor">Editor Staff answered 11 months ago

Right choice is (b) –\frac{7}{2}

Explanation: Given that: \vec{a}=\hat{i}-\hat{j}+3\hat{k}, \,\vec{b}=5\hat{i}-2\hat{j}+\hat{k} \,and \,\vec{c}=\hat{i}-\hat{j}

Also given, \vec{a}+μ\vec{b} is perpendicular to \vec{c}

Therefore, (\vec{a}+μ\vec{b}).\vec{c}=0

i.e. (\hat{i}-\hat{j}+3\hat{k}+μ(5\hat{i}-2\hat{j}+\hat{k})).(\hat{i}-\hat{j})=0

((1+5μ) \,\hat{i}-(1+2μ) \,\hat{j}+(μ+3) \,\hat{k}).(\hat{i}-\hat{j})=0

1+5μ+1+2μ=0

μ=-\frac{7}{2}.