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]

[N/A]