The angle between the vector a = i − 2j + 3k and the vector b = 3i − 2j + mk is 30° .

Find the values of m.

${\boldsymbol{a}} \cdot {\boldsymbol{b}} = \left| {\boldsymbol{a}} \right|\left| {\boldsymbol{b}} \right|\cos \theta$     (M1)

${\boldsymbol{a}} \cdot {\boldsymbol{b}} = \left( {\begin{array}{*{20}{c}}   1 \\   { - 2} \\   3 \end{array}} \right) \cdot \left( {\begin{array}{*{20}{c}}   3 \\   { - 2} \\   m \end{array}} \right) = 7 + 3m$     A1

$\left| {\boldsymbol{a}} \right| = \sqrt {14}$     $\left| {\boldsymbol{b}} \right| = \sqrt {13 + {m^2}}$     A1

$\left| {\boldsymbol{a}} \right|\left| {\boldsymbol{b}} \right|\cos \theta  = \sqrt {14} \sqrt {13 + {m^2}} \cos 30^\circ$

$7 + 3m = \sqrt {14} \sqrt {13 + {m^2}} \cos 30^\circ$     A1

m = 2.27, m = 25.7     A1A1

[6 marks]

Many candidates gained the first 4 marks by obtaining the equation, in unsimplified form, satisfied by m but then made mistakes in simplifying and solving this equation.