SPS SPS FM 2024 February — Question 4 8 marks

Exam BoardSPS
ModuleSPS FM (SPS FM)
Year2024
SessionFebruary
Marks8
TopicVectors 3D & Lines
TypeShow lines intersect and find intersection point
DifficultyStandard +0.3 This is a standard line intersection problem in 3D vector form. Students need to equate the two line equations, solve the resulting system of linear equations for the parameters, and verify the lines actually meet. While it requires careful algebraic manipulation and checking consistency across all three components, it's a routine Further Maths topic with well-practiced techniques and no novel insight required.
Spec4.04a Line equations: 2D and 3D, cartesian and vector forms4.04c Scalar product: calculate and use for angles4.04e Line intersections: parallel, skew, or intersecting
4. Two lines, \(l _ { 1 }\) and \(l _ { 2 }\), have the following equations. $$\begin{aligned} & l _ { 1 } : \mathbf { r } = \left( \begin{array} { c } - 11 \\ 10 \\ 3 \end{array} \right) + \lambda \left( \begin{array} { c } 2 \\ - 2 \\ 1 \end{array} \right) \\ & l _ { 2 } : \mathbf { r } = \left( \begin{array} { l } 5 \\ 2 \\ 4 \end{array} \right) \end{aligned}$$ 
4. Two lines, $l _ { 1 }$ and $l _ { 2 }$, have the following equations.

$$\begin{aligned}
& l _ { 1 } : \mathbf { r } = \left( \begin{array} { c } 
- 11 \\
10 \\
3
\end{array} \right) + \lambda \left( \begin{array} { c } 
2 \\
- 2 \\
1
\end{array} \right) \\
& l _ { 2 } : \mathbf { r } = \left( \begin{array} { l } 
5 \\
2 \\
4
\end{array} \right)
\end{aligned}$$

\hfill \mbox{\textit{SPS SPS FM 2024 Q4 [8]}}
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