Consider the two planes

     ${\pi _1}:4x + 2y - z = 8$

     ${\pi _2}:x + 3y + 3z = 3$.

Find the angle between ${\pi _1}$ and ${\pi _2}$, giving your answer correct to the nearest degree.

${{{n}}_1} = \left( {\begin{array}{*{20}{c}} 4 \\ 2 \\ { - 1} \end{array}} \right)\;\;\;{\text{and}}\;\;\;{{{n}}_2} = \left( {\begin{array}{*{20}{c}} 1 \\ 3 \\ 3 \end{array}} \right)$     (A1)(A1)

use of $\cos \theta  = \frac{{{{{n}}_1} \bullet {{{n}}_2}}}{{\left| {{{{n}}_1}} \right|\left| {{{{n}}_2}} \right|}}$     (M1)

$\cos \theta  = \frac{7}{{\sqrt {21} \sqrt {19} }}\;\;\;\left( { = \frac{7}{{\sqrt {399} }}} \right)$     (A1)(A1)

Note:     Award A1 for a correct numerator and A1 for a correct denominator.

$\theta  = 69^\circ$     A1

Note:     Award A1 for 111°.

[6 marks]

Reasonably well answered. A large number of candidates did not express their final answer correct to the nearest degree.