19N.1.AHL.TZ0.H_8

pestleMathematicsAAHLPaper 119N· ahl-3-14-vector-equation-of-linesource ↗

A straight line, ${L_\theta }$ , has vector equation r $= \left( {\begin{array}{*{20}{c}} 5 \\ 0 \\ 0 \end{array}} \right) + \lambda \left( {\begin{array}{*{20}{c}} 5 \\ {{\text{sin}}\,\theta } \\ {{\text{cos}}\,\theta } \end{array}} \right){\text{, }}\lambda {\text{, }}\theta \in \mathbb{R}$ .

The plane ${\Pi _p}$ , has equation $x = p{\text{, }}p \in \mathbb{R}$ .

Show that the angle between ${L_\theta }$ and ${\Pi _p}$ is independent of both $\theta$ and $p$ .

Markscheme / solution

a vector normal to ${\Pi _p}$ is $\left( {\begin{array}{*{20}{c}} 1 \\ 0 \\ 0 \end{array}} \right)$ (A1)

Note: Allow any scalar multiple of $\left( {\begin{array}{*{20}{c}} 1 \\ 0 \\ 0 \end{array}} \right)$ , including $\left( {\begin{array}{*{20}{c}} p \\ 0 \\ 0 \end{array}} \right)$

attempt to find scalar product (or vector product) of direction vector of line with any scalar multiple of $\left( {\begin{array}{*{20}{c}} 1 \\ 0 \\ 0 \end{array}} \right)$ M1

$\left( {\begin{array}{*{20}{c}} 1 \\ 0 \\ 0 \end{array}} \right) \bullet \left( {\begin{array}{*{20}{c}} 5 \\ {{\text{sin}}\,\theta } \\ {{\text{cos}}\,\theta } \end{array}} \right) = 5$ (or $\left( {\begin{array}{*{20}{c}} 1 \\ 0 \\ 0 \end{array}} \right) \times \left( {\begin{array}{*{20}{c}} 5 \\ {{\text{sin}}\,\theta } \\ {{\text{cos}}\,\theta } \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 0 \\ { - {\text{cos}}\,\theta } \\ {{\text{sin}}\,\theta } \end{array}} \right)$ ) A1

(if $\alpha$ is the angle between the line and the normal to the plane)

${\text{cos}}\,\alpha = \frac{5}{{1 \times \sqrt {25 + {\text{si}}{{\text{n}}^2}\,\theta + {\text{co}}{{\text{s}}^2}\,\theta } }}$ (or ${\text{sin}}\,\alpha = \frac{1}{{1 \times \sqrt {25 + {\text{si}}{{\text{n}}^2}\,\theta + {\text{co}}{{\text{s}}^2}\,\theta } }}$ ) A1

$\Rightarrow {\text{cos}}\,\alpha = \frac{5}{{\sqrt {26} }}$ or ${\text{sin}}\,\alpha = \frac{1}{{\sqrt {26} }}$ A1

this is independent of $p$ and $\theta$ , hence the angle between the line and the plane, $\left( {90 - \alpha } \right)$ , is also independent of $p$ and $\theta$ R1

Note: The final R mark is independent, but is conditional on the candidate obtaining a value independent of $p$ and $\theta$ .

[6 marks]

Examiners’ report

[N/A]