18M.1.SL.TZ2.S_1

pestleMathematicsAASLPaper 118M· ahl-3-13-scalar-(dot)-productsource ↗

Let $\mathop {{\text{OA}}}\limits^ \to = \left( \begin{gathered} 2 \hfill \\ 1 \hfill \\ 3 \hfill \\ \end{gathered} \right)$ and $\mathop {{\text{AB}}}\limits^ \to = \left( \begin{gathered} 1 \hfill \\ 3 \hfill \\ 1 \hfill \\ \end{gathered} \right)$ , where O is the origin. L₁ is the line that passes through A and B.

Find a vector equation for L₁.

[2]

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.

[3]

Markscheme / solution

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

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]

Examiners’ report

[N/A]