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

Two cables follow the lines r = (2î - ĵ + k̂) + λ(î + 2ĵ + 2k̂) and r = (-î + 2ĵ + k̂) + µ(î + ĵ)…

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

Two cables follow the lines $\vec{r} = (2\hat{i} - \hat{j} + \hat{k}) + \lambda(\hat{i} + 2\hat{j} + 2\hat{k})$ and $\vec{r} = (-\hat{i} + 2\hat{j} + \hat{k}) + \mu(\hat{i} + \hat{j})$ , with lengths in metres. Find the length of $\vec{b}_1 \times \vec{b}_2$ and the shortest distance between the cables.

Need a reminder? The key ideaShow

For skew lines, $d = \dfrac{\lvert (\vec{a}_2 - \vec{a}_1) \cdot (\vec{b}_1 \times \vec{b}_2) \rvert}{\lvert \vec{b}_1 \times \vec{b}_2 \rvert}$ . For parallel lines, $d = \dfrac{\lvert (\vec{a}_2 - \vec{a}_1) \times \vec{b} \rvert}{\lvert \vec{b} \rvert}$ . The common perpendicular has direction $\vec{b}_1 \times \vec{b}_2$ .

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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