IB Maths AA HL AA 3.4 Vectors 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, no calculator Paper 2, calculator

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Question 1

Paper 2, calculatorMedium6 marks

Figure 1 shows vectors $\mathbf u$ and $\mathbf v$ each drawn from the origin $O$

(a)

Find $\mathbf u\cdot\mathbf v$ and hence determine the angle between $\mathbf u$ and $\mathbf v$ in degrees correct to three significant figures.

[4]

(b)

The vector $\mathbf w=\mathbf u+k\mathbf v$ where $k\in\mathbb R$ Determine the value of $k$ for which $\mathbf w$ is perpendicular to $\mathbf u$

[2]

Question 2

Paper 2, calculatorHard6 marks

For $x,y,z,k\in\mathbb{R}$ consider the planes

P_1:\ x-z=1,\qquad P_2:\ y+z=2,\qquad Q_k:\ x+ky+(1-k)z=3.

(a)

Determine a direction vector of the line of intersection of $P_1$ and $P_2$

[1]

(b)

The acute angle between this line and $Q_k$ is $30^\circ$ Determine all possible values of $k$ and, for each value, the common solution of the three plane equations.

[5]

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