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

A straight cable runs along the line through A(0, 1, 2) with direction ratios (2, 2, 1)…

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

A straight cable runs along the line through $A(0, 1, 2)$ with direction ratios $(2, 2, 1)$ . A sensor sits at $P(6, 4, 2)$ , and all lengths are in metres. A technician needs to run a wire from the sensor to the nearest point of the cable. Find the parameter $t$ of that point, the coordinates of the point, and the length of the wire.

Need a reminder? The key ideaShow

The foot of the perpendicular from $P$ to the line $\vec{r} = \vec{a} + t\vec{b}$ has $t = \dfrac{(P - \vec{a}) \cdot \vec{b}}{\lvert \vec{b} \rvert^2}$ . The distance is $PF$ , or $\dfrac{\lvert (P - \vec{a}) \times \vec{b} \rvert}{\lvert \vec{b} \rvert}$ .

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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