Pre-U Pre-U 9794/2 2013 November — Question 9

Exam BoardPre-U
ModulePre-U 9794/2 (Pre-U Mathematics Paper 2)
Year2013
SessionNovember
TopicVectors 3D & Lines
TypeLine intersection verification
DifficultyModerate -0.3 This is a standard textbook exercise on line intersection requiring systematic equation setup and solving a 3×2 system. While it involves multiple steps (equating components, solving for parameters, verification), the method is routine and well-practiced. Slightly easier than average because it's guaranteed to intersect (no need to handle the non-intersection case) and the arithmetic is manageable.
Spec1.10a Vectors in 2D: i,j notation and column vectors1.10b Vectors in 3D: i,j,k notation4.04a Line equations: 2D and 3D, cartesian and vector forms

9 Two straight lines have equations $$\mathbf { r } = \left( \begin{array} { r } 16 \\ 2 \\ 3 \end{array} \right) + \lambda \left( \begin{array} { r } 3 \\ 2 \\ - 1 \end{array} \right) \quad \text { and } \quad \mathbf { r } = \left( \begin{array} { r } - 3 \\ 8 \\ 12 \end{array} \right) + \mu \left( \begin{array} { r } 5 \\ - 6 \\ - 3 \end{array} \right) .$$ Show that the two lines intersect and find the coordinates of their point of intersection.

9 Two straight lines have equations

$$\mathbf { r } = \left( \begin{array} { r } 
16 \\
2 \\
3
\end{array} \right) + \lambda \left( \begin{array} { r } 
3 \\
2 \\
- 1
\end{array} \right) \quad \text { and } \quad \mathbf { r } = \left( \begin{array} { r } 
- 3 \\
8 \\
12
\end{array} \right) + \mu \left( \begin{array} { r } 
5 \\
- 6 \\
- 3
\end{array} \right) .$$

Show that the two lines intersect and find the coordinates of their point of intersection.

\hfill \mbox{\textit{Pre-U Pre-U 9794/2 2013 Q9}}