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

A lighthouse stands at the origin of a map, with distances in kilometres…

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

A lighthouse stands at the origin of a map, with distances in kilometres. At noon a ship is at the point $(-25, 0)$ and sails in a straight line at the constant velocity $\vec v = (3, 4)$ kilometres per hour. The captain must report the closest distance between the ship and the lighthouse, and when it happens. After $t$ hours the position of the ship is $\vec r = (-25, 0) + t\vec v$ . Find the time of closest approach, the closest distance and the speed of the ship.

Need a reminder? The key ideaShow

The scalar projection of $\vec a$ on $\vec b$ is $\dfrac{\vec a\cdot\vec b}{\lvert\vec b\rvert}$ , and $\lvert\vec a\rvert^2 = (\text{projection})^2 + \lvert\vec a_{\perp}\rvert^2$ . For a constant force, the work is $W = \vec F\cdot\vec d$ .

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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