The set of real values of \(\lambda\) for which the vectors \[ \lambda \hat{i}-3\hat{j}+5\hat{k} \] and \[ 2\lambda \hat{i}-\lambda \hat{j}+\hat{k} \] are perpendicular to each other is

Question

Let the arc AC of a circle subtend a right angle at the center O. If the point B on the arc AC divides the arc AC such that: \[ \frac{\text{length of arc AB}}{\text{length of arc BC}} = \frac{1}{5} \] and \[ \overrightarrow{OC} = \alpha \overrightarrow{OA} + \beta \overrightarrow{OB}, \] then \( \alpha = \sqrt{2} (\sqrt{3}-1) \beta \) is equal to:

Answer 1

Step 1: Set up the dot product equation.
Two vectors are perpendicular when their dot product is zero. \[(\lambda)(2\lambda) + (-3)(-\lambda) + (5)(1) = 0\] \[2\lambda^2 + 3\lambda + 5 = 0.\]

Step 2: Check the discriminant.
Discriminant \(= 9 - 40 = -31 < 0\). No real solutions exist.
\[\boxed{\phi \text{ (empty set)}}\]

The set of real values of \(\lambda\) for which the vectors \[ \lambda \hat{i}-3\hat{j}+5\hat{k} \] and \[ 2\lambda \hat{i}-\lambda \hat{j}+\hat{k} \] are perpendicular to each other is

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The Correct Option is D

Solution and Explanation

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