(i)     Find $\overrightarrow {{\text{OA}}}  \times \overrightarrow {{\text{OB}}}$ .

(ii)     Determine the area of the triangle OAB.

(iii)     Find the Cartesian equation of the plane OAB.

(i)     Find the vector equation of the line ${L_1}$ containing the points A and B.

(ii)     The line ${L_2}$ has vector equation $\left( {\begin{array}{*{20}{c}}   x \\   y \\   z \end{array}} \right) = \left( {\begin{array}{*{20}{c}}   2 \\   4 \\   3 \end{array}} \right) + \mu \left( {\begin{array}{*{20}{c}}   1 \\   3 \\   2 \end{array}} \right)$.

Determine whether or not ${L_1}$ and ${L_2}$ are skew.

(i)     $\overrightarrow {{\text{OA}}} \times \overrightarrow {{\text{OB}}} =$ i + 7j – 5k     A1

(ii)     area $= \frac{1}{2}|$i + 7j – 5k$| = \frac{{5\sqrt 3 }}{2}{\text{(4.33)}}$     M1A1

(iii)     equation of plane is $x + 7y - 5z = k$     M1

$x + 7y - 5z = 0$     A1

[5 marks]

(i)     direction of line = (3i + j + 2k) – (i + 2j + 3k) = 2i – j – k     M1A1

equation of line is

r = (i + 2j + 3k) + $\lambda$(2i – j – k)     A1

(ii)     at a point of intersection,

$1 + 2\lambda = 2 + \mu$

$2 - \lambda = 4 + 3\mu$     M1A1

$3 - \lambda = 3 + 2\mu$

solving the ${2^{{\text{nd}}}}$ and ${3^{{\text{rd}}}}$ equations, $\lambda = 4{\text{, }}\mu = - 2$     A1

these values do not satisfy the ${1^{{\text{st}}}}$ equation so the lines are skew     R1 

[7 marks]