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

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Question 1

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 2

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]

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