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

A drone takes off from a launch pad at the origin…

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

A drone takes off from a launch pad at the origin. After a short flight along a straight line it is at the point with coordinates $(30, 40, 120)$ metres. The $x$ -axis points east, the $y$ -axis points north and the $z$ -axis points up. Write its displacement as a vector, and find how far the drone is from the pad. Then find the angle that its path makes with the vertical.

Need a reminder? The key ideaShow

A vector $x\hat i + y\hat j + z\hat k$ has magnitude $\sqrt{x^2 + y^2 + z^2}$ and unit vector $\hat a = \dfrac{\vec a}{\lvert\vec a\rvert}$ . Its direction cosines are $l = \dfrac{x}{\lvert\vec a\rvert}$ , $m = \dfrac{y}{\lvert\vec a\rvert}$ , $n = \dfrac{z}{\lvert\vec a\rvert}$ , and $l^2 + m^2 + n^2 = 1$ .

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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