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Date May 2018 Marks available 3 Reference code 18M.1.sl.TZ2.1
Level SL only Paper 1 Time zone TZ2
Command term Find Question number 1 Adapted from N/A

Question

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]
a.

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]
b.

Markscheme

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]

a.

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]

b.

Examiners report

[N/A]
a.
[N/A]
b.

Syllabus sections

Topic 4 - Vectors » 4.3 » Vector equation of a line in two and three dimensions: \(r = a + tb\) .
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