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IB Mathematics AHL 3.15- Different types of lines lines AA HL Paper 3 | Exam Style Questions

IB Mathematics AHL 3.15- Different types of lines lines AA HL Paper 3- Exam Style Questions

IB DP Math AA: HL Papers 3: All Topics
Question

Consider the lines \( L_1 \), \( L_2 \), \( L_3 \), and \( L_4 \), with respective equations:

\[ L_1: \left( \begin{array}{c} x \\ y \\ z \end{array} \right) = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} + t \begin{pmatrix} 3 \\ -2 \\ 1 \end{pmatrix} \]

\[ L_2: \left( \begin{array}{c} x \\ y \\ z \end{array} \right) = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} + p \begin{pmatrix} 3 \\ 2 \\ 1 \end{pmatrix} \]

\[ L_3: \left( \begin{array}{c} x \\ y \\ z \end{array} \right) = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} + s \begin{pmatrix} -1 \\ 2 \\ -a \end{pmatrix} \]

\[ L_4: \left( \begin{array}{c} x \\ y \\ z \end{array} \right) = q \begin{pmatrix} -6 \\ 4 \\ -2 \end{pmatrix} \]

(a) Write down a line that is parallel to \( L_4 \).[1]

(b) Write down the position vector of the point of intersection of \( L_1 \) and \( L_2 \).[1]

(c) Given that \( L_1 \) is perpendicular to \( L_3 \), find the value of a.[5]

▶️ Answer/Explanation
Markscheme Solution

(a)

\[ \left( \begin{array}{c} x \\ y \\ z \end{array} \right) = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix} + t \begin{pmatrix} -6 \\ 4 \\ -2 \end{pmatrix} \] (A1)

Note: Any line with a direction vector proportional to \( \begin{pmatrix} -6 \\ 4 \\ -2 \end{pmatrix} \) is valid. [1 mark]

(b)

\[ \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} \] (A1) [1 mark]

(c)

Direction vectors: \( \begin{pmatrix} 3 \\ -2 \\ 1 \end{pmatrix} \), \( \begin{pmatrix} -1 \\ 2 \\ -a \end{pmatrix} \) (A1)(A1)

Since \( L_1 \perp L_3 \), their direction vectors satisfy \( \mathbf{a} \cdot \mathbf{b} = 0 \): (M1)

\[ (3)(-1) + (-2)(2) + (1)(-a) = -3 – 4 – a = 0 \] (A1)

\[ a = -7 \] (A1) [5 marks]

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