For which values of λ is the angle between λ i + 2 j + k and λ i - 3 j + 2 k obtuse?

Question 89

HardOriginal question

For which values of $\lambda$ is the angle between $\lambda\hat{i} + 2\hat{j} + \hat{k}$ and $\lambda\hat{i} - 3\hat{j} + 2\hat{k}$ obtuse?

\lambda < -2

\lambda > 2

Correct answerYour answer

0 < \lambda < 2

Correct answerYour answer

-4 < \lambda < 4

Correct answerYour answer

-2 < \lambda < 2

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer D

Correct answer: D

The angle is obtuse when $\vec{a} \cdot \vec{b} < 0$ ; these vectors are never opposite, so $\theta = \pi$ does not arise.
$\vec{a} \cdot \vec{b} = \lambda^{2} - 6 + 2 = \lambda^{2} - 4 < 0$ gives $-2 < \lambda < 2$ .

Worth noting

$\lambda < -2$ or $\lambda > 2$ makes the scalar product positive: that is when the angle is acute.

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