Six equilateral triangles, each with side length 3 cm, are arranged to form a hexagon.
This is shown in the following diagram.

The vectors p , q and r are shown on the diagram.

Find p•(p + q + r).

METHOD 1 (using |p| |2q| cosθ)

finding p + q + r      (A1)

eg  2q, 

| p + q + r | = 2 × 3 (= 6)  (seen anywhere)     A1

correct angle between p and q (seen anywhere)      (A1)

$\frac{\pi }{3}$  (accept 60°)

substitution of their values     (M1)

eg  3 × 6 × cos$\left( {\frac{\pi }{3}} \right)$

correct value for cos$\left( {\frac{\pi }{3}} \right)$ (seen anywhere)     (A1)

eg  $\frac{1}{2},\,\,\,3 \times 6 \times \frac{1}{2}$

p•(p + q + r) = 9     A1 N3

METHOD 2 (scalar product using distributive law)

correct expression for scalar distribution      (A1)

eg  p• p + p•q + p•r

three correct angles between the vector pairs (seen anywhere)      (A2)

eg  0° between p and p, $\frac{\pi }{3}$ between p and q, $\frac{{2\pi }}{3}$ between p and r

Note: Award A1 for only two correct angles.

substitution of their values      (M1)

eg  3.3.cos0 +3.3.cos$\frac{\pi }{3}$ + 3.3.cos120

one correct value for cos0, cos$\left( {\frac{\pi }{3}} \right)$ or cos$\left( {\frac{2\pi }{3}} \right)$ (seen anywhere)      A1

eg  $\frac{1}{2},\,\,\,3 \times 6 \times \frac{1}{2}$

p•(p + q + r) = 9     A1 N3

METHOD 3 (scalar product using relative position vectors)

valid attempt to find one component of p or r      (M1)

eg   sin 60 = $\frac{x}{3}$, cos 60 = $\frac{x}{3}$, one correct value $\frac{3}{2},\,\,\frac{{3\sqrt 3 }}{2},\,\,\frac{{ - 3\sqrt 3 }}{2}$

one correct vector (two or three dimensions) (seen anywhere)      A1

eg  $p = \left( \begin{gathered} \,\,\,\frac{3}{2} \hfill \\ \frac{{3\sqrt 3 }}{2} \hfill \\ \end{gathered} \right),\,\,q = \left( \begin{gathered} 3 \hfill \\ 0 \hfill \\ \end{gathered} \right),\,\,r = \left( \begin{gathered} \,\,\,\,\frac{3}{2} \hfill \\ - \frac{{3\sqrt 3 }}{2} \hfill \\ \,\,\,\,0 \hfill \\ \end{gathered} \right)$

three correct vectors p + q + r = 2q     (A1)

p + q + r = $\left( \begin{gathered} 6 \hfill \\ 0 \hfill \\ \end{gathered} \right)$ or $\left( \begin{gathered} 6 \hfill \\ 0 \hfill \\ 0 \hfill \\ \end{gathered} \right)$ (seen anywhere, including scalar product)      (A1)

correct working       (A1)
eg  $\left( {\frac{3}{2} \times 6} \right) + \left( {\frac{{3\sqrt 3 }}{2} \times 0} \right),\,\,9 + 0 + 0$

p•(p + q + r) = 9     A1 N3

[6 marks]