Find a vector equation for L₁.

The vector \(\left( \begin{gathered}
2 \hfill \\
p \hfill \\
0 \hfill \\
\end{gathered} \right)\) is perpendicular to \(\mathop {{\text{AB}}}\limits^ \to  \). Find the value of p.

any correct equation in the form r = a + tb (accept any parameter for t)

where a is \(\left( \begin{gathered}
2 \hfill \\
1 \hfill \\
3 \hfill \\
\end{gathered} \right)\), and b is a scalar multiple of \(\left( \begin{gathered}
1 \hfill \\
3 \hfill \\
1 \hfill \\
\end{gathered} \right)\)     A2 N2

eg r = \(\left( \begin{gathered}
2 \hfill \\
1 \hfill \\
3 \hfill \\
\end{gathered} \right) = t\left( \begin{gathered}
1 \hfill \\
3 \hfill \\
1 \hfill \\
\end{gathered} \right)\), r = 2i + j + 3k + s(i + 3j + k)

Note: Award A1 for the form a + tb, A1 for the form L = a + tb, A0 for the form r = b + ta.

[2 marks]

METHOD 1

correct scalar product     (A1)

eg  (1 × 2) + (3 × p) + (1 × 0), 2 + 3p

evidence of equating their scalar product to zero     (M1)

eg  a•b = 0, 2 + 3p = 0, 3p = −2

\(p =  - \frac{2}{3}\)       A1 N3

METHOD 2

valid attempt to find angle between vectors      (M1)

correct substitution into numerator and/or angle       (A1)

eg  \({\text{cos}}\,\theta  = \frac{{\left( {1 \times 2} \right) + \left( {3 \times p} \right) + \left( {1 \times 0} \right)}}{{\left| a \right|\left| b \right|}},\,\,{\text{cos}}\,\theta  = 0\)

\(p =  - \frac{2}{3}\)       A1 N3

[3 marks]