Let a and b be two unit vectors such that the angle between them is 3 π . If λ a + 2 b and 3 a − λ b are perpendicular to each other, then the number of values of λ in [ − 1 , 3 ] is :

a ^ ⋅ b ^ = 2 1 Now ( λ a ^ + 2 b ^ ) ⋅ ( 3 a ^ − λ b ^ ) = 0 3 λ a ^ ⋅ a ^ − λ 2 a ^ ⋅ b ^ + 6 a ^ ⋅ b ^ − 2 λ b ^ ⋅ b ^ = 0 3 λ − 2 λ 2 + 3 − 2 λ = 0 λ 2 − 2 λ − 6 = 0 λ = 1 ± 7 ⇒ number of values = 0

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MathematicsVectorGeneral definitions, Angle between vectors, Section formula, Geometrical resultsMedium2 minPYQ_2025

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Let $a$ and $b$ be two unit vectors such that the angle between them is $\frac{π}{3}$ . If $λ a + 2 b$ and $3 a - λ b$ are perpendicular to each other, then the number of values of $λ$ in $[- 1, 3]$ is :

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Solution:

$\overset{a}{^} \cdot \hat{b} = \frac{1}{2} Now (λ \overset{a}{^} + 2 \hat{b}) \cdot (3 \overset{a}{^} - λ \hat{b}) = 0 3 λ \overset{a}{^} \cdot \overset{a}{^} - λ^{2} \overset{a}{^} \cdot \hat{b} + 6 \overset{a}{^} \cdot \hat{b} - 2 λ \hat{b} \cdot \hat{b} = 0 3 λ - \frac{λ ^{2}}{2} + 3 - 2 λ = 0 λ^{2} - 2 λ - 6 = 0 λ = 1 \pm 7 \Rightarrow number of values = 0$

Stream:JEESubject:MathematicsTopic:VectorSubtopic:General definitions, Angle between vectors, Section formula, Geometrical results

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