a and b are unit vectors at an angle θ. What is | a - b |?

Question 88

HardOriginal question

$\vec{a}$ and $\vec{b}$ are unit vectors at an angle $\theta$ . What is $\lvert \vec{a} - \vec{b} \rvert$ ?

2\cos\frac{\theta}{2}

Correct answerYour answer

2 - 2\cos \theta

Correct answerYour answer

\sqrt{2}\sin\frac{\theta}{2}

Correct answerYour answer

2\sin\frac{\theta}{2}

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer D

Correct answer: D

$\lvert \vec{a} - \vec{b} \rvert^{2} = 1 + 1 - 2\cos \theta = 2(1 - \cos \theta)$ .
$1 - \cos \theta = 2\sin^{2}\frac{\theta}{2}$ , so $\lvert \vec{a} - \vec{b} \rvert = 2\sin\frac{\theta}{2}$ , which is not negative since $\frac{\theta}{2}$ lies in $\left[0, \frac{\pi}{2}\right]$ .

Worth noting

$2\cos\frac{\theta}{2}$ is $\lvert \vec{a} + \vec{b} \rvert$ , where $2\cos \theta$ is added, not taken away.

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