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A surveyor records two displacement vectors a and b with lvert arvert = 3 metres and lvert brvert =…

Try it on paper first. A hint, the key idea and the worked solution are here when you need them.

A surveyor records two displacement vectors $\vec a$ and $\vec b$ with $\lvert\vec a\rvert = 3$ metres and $\lvert\vec b\rvert = 2$ metres. She notices that the vector $\vec a + 2\vec b$ is perpendicular to the vector $\vec a - \vec b$ , but she has lost the angle between $\vec a$ and $\vec b$ . Find $\vec a\cdot\vec b$ , and then find the angle between $\vec a$ and $\vec b$ in degrees.

Need a reminder? The key ideaShow

$\vec a\cdot\vec b = \lvert\vec a\rvert\lvert\vec b\rvert\cos\theta$ , and $\lvert\vec a \pm \vec b\rvert^2 = \lvert\vec a\rvert^2 \pm 2\,\vec a\cdot\vec b + \lvert\vec b\rvert^2$ . The dot product follows the usual rules of expanding brackets, with $\vec a\cdot\vec a = \lvert\vec a\rvert^2$ .

The worked solution

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Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.

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