16N.1.AHL.TZ0.H_4

pestleMathematicsAAHLPaper 116N· ahl-3-16-vector-productsource ↗

Consider the vectors a $=$ i $- {\text{ }}3$ j $- {\text{ }}2$ k, b $= - {\text{ }}3$ j $+ {\text{ }}2$ k.

Find a $\times$ b.

[2]

Hence find the Cartesian equation of the plane containing the vectors a and b, and passing through the point $(1,{\text{ }}0,{\text{ }} - 1)$ .

[3]

Markscheme / solution

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

a $\times$ b $= - 12$ i $- {\text{ }}2$ j $- {\text{ }}3$ k (M1)A1

[2 marks]

METHOD 1

$- 12x - 2y - 3z = d$ M1

$- 12 \times 1 - 2 \times 0 - 3( - 1) = d$ (M1)

$\Rightarrow d = - 9$ A1

$- 12x - 2y - 3z = - 9{\text{ }}({\text{or }}12x + 2y + 3z = 9)$

METHOD 2

$\left( {\begin{array}{*{20}{c}} x \\ y \\ z \end{array}} \right) \bullet \left( {\begin{array}{*{20}{c}} { - 12} \\ { - 2} \\ { - 3} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 1 \\ 0 \\ { - 1} \end{array}} \right) \bullet \left( {\begin{array}{*{20}{c}} { - 12} \\ { - 2} \\ { - 3} \end{array}} \right)$ M1A1

$- 12x - 2y - 3z = - 9{\text{ }}({\text{or }}12x + 2y + 3z = 9)$ A1

[3 marks]

Examiners’ report

[N/A]