IB Maths AI HL AI 3.3 Vectors and matrix transformations questions

Exam-style questions in the shape of Paper 1 and Paper 2, with full markschemes. Read the markscheme, or type, photograph or draw your answer and get it marked.

Paper

Paper 1 Paper 2

Difficulty

Easy Medium Hard

Question 1

Paper 1Medium6 marks

Two survey boats, A and B, move at constant velocities in a horizontal plane. Coordinates are measured in metres, with the $x$ -axis pointing east and the $y$ -axis pointing north. At $t=0$ their position vectors are

\mathbf r_A=(120,30), \qquad \mathbf r_B=(20,100),

and their velocity vectors are

\mathbf v_A=(3,1), \qquad \mathbf v_B=(5,-2),

in $\text{m s}^{-1}$ The time $t$ is measured in seconds.

(a)

Find an expression, in terms of $t$ for the relative position vector of B from A, $\overrightarrow{AB}$

[2]

(b)

Using your answer to part (a), determine the time $t\geq 0$ at which the boats are closest and their minimum separation. Give both answers to 3 significant figures.

[4]

Question 2

Paper 2Medium7 marks

A triangular inspection plate is fitted in a three-dimensional frame. From vertex $O$ the edges $OA$ and $OB$ are represented by the vectors

\mathbf a=(2,1,2)\text{ m},\qquad \mathbf b=(1,3,-1)\text{ m}.

An engineer uses these edges to assess the size and orientation of the plate.

(a)

Determine the area of triangle $OAB$

[2]

(b)

Determine the acute angle between the lines $OA$ and $OB$ to the nearest $0.1^\circ$

[2]

(c)

Find the scalar component of $\mathbf a$ in the direction of $\mathbf b$ and the magnitude of the component of $\mathbf a$ perpendicular to $\mathbf b$ Give each answer to 3 significant figures.

[2]

(d)

Use your answers to part (c) to state whether $OA$ is more nearly parallel or perpendicular to $OB$ giving a reason.

[1]

Question 3

Paper 2Medium6 marks

For the first 10 hours of a survey, two vessels move in a horizontal plane at constant velocity. Their position vectors, relative to a fixed origin, are

\mathbf r_A(t)=(1.5,\,2.0)+t(0.40,\,0.30),\qquad \mathbf r_B(t)=(6.7,\,5.1)+t(-0.35,\,-0.10),

where coordinates are in kilometres and $t$ is the time in hours after 08:00.

(a)

Write down the initial position vector of B relative to A, $\overrightarrow{AB}(0)$

[1]

(b)

Show that

\overrightarrow{AB}(t)=(5.2,\,3.1)+t(-0.75,\,-0.40).

[2]

(c)

Hence determine the time, $t$ and the distance between the vessels when they are closest during the 10-hour survey. Give both answers to 3 significant figures.

[3]

The best questions are still waiting

Unlock every question in this topic, then type, photograph or draw your answer and have it marked against the markscheme. Find out where you are dropping marks.

Unlock all questions Compare Student and Student Plus