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

A survey drone flies along the line r = (-2î + ĵ + 4k̂) + λ(3î + 2ĵ - k̂)…

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

A survey drone flies along the line $\vec{r} = (-2\hat{i} + \hat{j} + 4\hat{k}) + \lambda(3\hat{i} + 2\hat{j} - \hat{k})$ . Coordinates are in hundreds of metres. Find the coordinates of the drone when $\lambda = 2$ . Then find the value of $\lambda$ for which the drone is at the point $(13, 11, -1)$ , which is where the landing marker is placed.

Need a reminder? The key ideaShow

A line through $\vec{a}$ parallel to $\vec{b}$ is $\vec{r} = \vec{a} + \lambda\vec{b}$ . In Cartesian form it is $\dfrac{x - x_1}{a} = \dfrac{y - y_1}{b} = \dfrac{z - z_1}{c}$ . A point lies on the line if one value of $\lambda$ fits all three coordinates.

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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← A surveyor fixes a laser at the origin of a hall, with the axes along two walls and the vertical…Two survey markers stand at the points A(2, -3, 1) and B(8, 1, -3), with coordinates in metres… →

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