EasyMathematicsHigh School (11–12)Vectors and Three-Dimensional GeometryNumerical

Two steel rods are welded at a joint. Treat them as the vectors a and b drawn from the joint…

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

Two steel rods are welded at a joint. Treat them as the vectors $\vec a$ and $\vec b$ drawn from the joint. The rod $\vec a$ is $3$ metres long, the rod $\vec b$ is $5$ metres long, and the angle between them is $60^\circ$ . Find $\vec a\cdot\vec b$ . Then complete the parallelogram on the two rods and find the lengths of its two diagonals, $\lvert\vec a + \vec b\rvert$ and $\lvert\vec a - \vec b\rvert$ .

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

Have you tried it?

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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