16N.2.AHL.TZ0.H_2

pestleMathematicsAAHLPaper 216N· ahl-3-18-intersections-of-lines-&-planessource ↗

Find the acute angle between the planes with equations $x + y + z = 3$ and $2x - z = 2$ .

Markscheme / solution

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

n $_1 = \left( {\begin{array}{*{20}{c}} 1 \\ 1 \\ 1 \end{array}} \right)$ and n $_2 = \left( {\begin{array}{*{20}{c}} 2 \\ 0 \\ { - 1} \end{array}} \right)$ (A1)(A1)

EITHER

$\theta = \arccos \left( {\frac{{{n_1} \bullet {n_2}}}{{\left| {{n_1}} \right|\left| {{n_2}} \right|}}} \right)\left( {\cos \theta = \frac{{{n_1} \bullet {n_2}}}{{\left| {{n_1}} \right|\left| {{n_2}} \right|}}} \right)$ (M1)

$= \arccos \left( {\frac{{2 + 0 - 1}}{{\sqrt 3 \sqrt 5 }}} \right)\left( {\cos \theta = \frac{{2 + 0 - 1}}{{\sqrt 3 \sqrt 5 }}} \right)$ (A1)

$= \arccos \left( {\frac{1}{{\sqrt {15} }}} \right)\left( {\cos \theta = \frac{1}{{\sqrt {15} }}} \right)$

$\theta = \arcsin \left( {\frac{{\left| {{n_1} \times {n_2}} \right|}}{{\left| {{n_1}} \right|\left| {{n_2}} \right|}}} \right)\left( {\sin \theta = \frac{{\left| {{n_1} \times {n_2}} \right|}}{{\left| {{n_1}} \right|\left| {{n_2}} \right|}}} \right)$ (M1)

$= \arcsin \left( {\frac{{\sqrt {14} }}{{\sqrt 3 \sqrt 5 }}} \right)\left( {\sin \theta = \frac{{\sqrt {14} }}{{\sqrt 3 \sqrt 5 }}} \right)$ (A1)

$= \arcsin \left( {\frac{{\sqrt {14} }}{{\sqrt {15} }}} \right)\left( {\sin \theta = \frac{{\sqrt {14} }}{{\sqrt {15} }}} \right)$

THEN

$= 75.0^\circ {\text{ (or 1.31)}}$ A1

[5 marks]

Examiners’ report

[N/A]