IB Maths AI HL 3.12 Vector kinematics Question Bank

Practise IB Mathematics HL 3.12 by applying vector kinematics methods to exam-style questions.

[Maximum number: 8]

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)

Write down the position vector, $\overrightarrow{\mathrm{OD}}$ , of the drone at time t.

[ 1 ]

$\overrightarrow{\mathrm{OD}}=\left(\begin{smallmatrix}200\\-100\\0.02\end{smallmatrix}\right)+t\left(\begin{smallmatrix}-15\\-20\\0\end{smallmatrix}\right)$

(b)

At time $t_1$ , the drone passes through the point with position vector $\left(\begin{smallmatrix}152\\p\\0.02\end{smallmatrix}\right)$ .
Find the value of

[ 3 ]

(i)

$t_{1}$

[ 2 ]

$\left(\begin{smallmatrix}200\\-100\\0.02\end{smallmatrix}\right)+t\left(\begin{smallmatrix}-15\\-20\\0\end{smallmatrix}\right)=\left(\begin{smallmatrix}152\\p\\0.02\end{smallmatrix}\right)$
OR 200-15t=152 $t_1=3.2\ (\frac{16}{5})$

(ii)

p.

[ 1 ]

$\quad p=-164$

(c)

Hence, find the shortest distance between the car and the drone.

[ 4 ]

Attempt to find $\left|\left(\begin{smallmatrix}190-65t\\-95+13t\\0.02\end{smallmatrix}\right)\right|$ .
Attempt to find a minimum, for example t=3.09 hours.
Closest distance $=55.9\ (55.8931\ldots)\,\mathrm{km}$ .

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AHL 3.12 (HL)—Vector kinematics question 1

Question (a)

Question (b)

Question (i)

Question (ii)

Question (c)