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Exam-Style Question on Vectors

A mathematics exam-style question with a worked solution that can be revealed gradually

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Question id: 69. This question is similar to one that appeared on an IB Standard paper in 2014. The use of a calculator is not allowed.

George and Hugo like to fly model airplanes. On one day George's plane takes off from level ground and shortly after that Hugo's plane takes off.

The position of George’s plane \(s\) seconds after it takes off is given by \(\begin{pmatrix} 1 \\ 2 \\ 0 \\ \end{pmatrix} + s\begin{pmatrix} 5 \\ -2 \\ 6 \\ \end{pmatrix} \) where the distances are in metres.

(a) Find the speed of George’s plane to the nearest integer.

(b) Find the height of George’s plane after four seconds.

The position of Hugo’s airplane \(t\) seconds after it takes off is given by \(\begin{pmatrix} 4 \\ -4 \\ 0 \\ \end{pmatrix}+t\begin{pmatrix} 7 \\ -2 \\ 9 \\ \end{pmatrix} \) where the distances are in metres.

(c) Show that the paths of the planes are not perpendicular.

The two airplanes collide at the point \((46, -16, 54)\).

(d) How long after George’s plane takes off does Hugo’s plane take off ?

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