IB Maths AI HL 3.10 Vectors Question Bank

Practise IB Mathematics HL 3.10 by applying vectors methods to exam-style questions.

[Maximum number: 4]

In this question, all distances are in kilometres and t is in hours.
Let $\left(\begin{smallmatrix}1\\0\\0\end{smallmatrix}\right)$ be a displacement of 1 km due east, $\left(\begin{smallmatrix}0\\1\\0\end{smallmatrix}\right)$ be a displacement of 1 km due north, and $\left(\begin{smallmatrix}0\\0\\1\end{smallmatrix}\right)$ be a vertical displacement of 1 km.
Highway 85 in Saudi Arabia is a long, straight, flat road.
Relative to the centre of the town Arar, point O, the position vector of a car, C, travelling along this road is given by:

\overrightarrow{\mathrm{OC}}=\left(\begin{array}{c}10\\-5\\0\end{array}\right)+t\left(\begin{array}{c}50\\-33\\0\end{array}\right)

The police are testing a long-range drone, D, to monitor cars travelling along this road. The drone is launched at t=0 from the point with position vector $\left(\begin{smallmatrix}200\\-100\\0.02\end{smallmatrix}\right)$ and flies in a straight line with a constant height of 0.02 km and a constant velocity of $\left(\begin{smallmatrix}-15\\-20\\0\end{smallmatrix}\right)$ .

(a)

Find the speed of the car.

[ 2 ]

Recognition that speed is the magnitude of $\left(\begin{smallmatrix}50\\-33\\0\end{smallmatrix}\right)$ .
$\sqrt{50^2+(-33)^2}$ OR $\left|\left(\begin{smallmatrix}50\\-33\\0\end{smallmatrix}\right)\right|$ $=59.9\,\mathrm{km\,h^{-1}}\ (59.9082\ldots)$

(b)

Find an expression for $\overrightarrow{\mathrm{CD}}$ , the relative position of the drone from the car.

[ 2 ]

Attempt to find the difference between the two position vectors.
$\left(\begin{smallmatrix}190-65t\\-95+13t\\0.02\end{smallmatrix}\right)$
Note: Award A1M1A0 for $\left(\begin{smallmatrix}-190+65t\\95-13t\\-0.02\end{smallmatrix}\right)$ .

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AHL 3.10 (HL)—Vectors question 1

Question (a)

Question (b)