Find a $\times$ b.

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

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]