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

Two cables hang in a stadium. The first has direction ratios (2, 3, 6) and the second has direction…

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

Two cables hang in a stadium. The first has direction ratios $(2, 3, 6)$ and the second has direction ratios $(1, -4, 8)$ . Find $\cos\theta$ for the acute angle between them, and the angle itself.

Need a reminder? The key ideaShow

The acute angle between lines with directions $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$ has $\cos\theta = \dfrac{\lvert a_1a_2 + b_1b_2 + c_1c_2 \rvert}{\sqrt{a_1^2 + b_1^2 + c_1^2}\,\sqrt{a_2^2 + b_2^2 + c_2^2}}$ . The lines are perpendicular when the dot product is $0$ .

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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