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Date None Specimen Marks available 5 Reference code SPNone.2.sl.TZ0.4
Level SL only Paper 2 Time zone TZ0
Command term Find Question number 4 Adapted from N/A

Question

Consider the lines \({L_1}\) , \({L_2}\) , \({L_2}\) , and \({L_4}\) , with respective equations.

\({L_1}\) : \(\left( {\begin{array}{*{20}{c}}
x\\
y\\
z
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
1\\
2\\
3
\end{array}} \right) + t\left( {\begin{array}{*{20}{c}}
3\\
{ - 2}\\
1
\end{array}} \right)\)

\({L_2}\)  : \(\left( \begin{array}{l}
x\\
y\\
z
\end{array} \right) = \left( \begin{array}{l}
1\\
2\\
3
\end{array} \right) + p\left( \begin{array}{l}
3\\
2\\
1
\end{array} \right)\)

\({L_3}\) : \(\left( {\begin{array}{*{20}{c}}
x\\
y\\
z
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
0\\
1\\
0
\end{array}} \right) + s\left( {\begin{array}{*{20}{c}}
{ - 1}\\
2\\
{ - a}
\end{array}} \right)\)

\({L_4}\) : \(\left( {\begin{array}{*{20}{c}}
x\\
y\\
z
\end{array}} \right) = q\left( {\begin{array}{*{20}{c}}
{ - 6}\\
4\\
{ - 2}
\end{array}} \right)\)

 

Write down the line that is parallel to \({L_4}\) .

[1]
a.

Write down the position vector of the point of intersection of \({L_1}\) and \({L_2}\) .

[1]
b.

Given that \({L_1}\) is perpendicular to \({L_3}\) , find the value of a .

[5]
c.

Markscheme

\({L_1}\)     A1     N1

[1 mark]

a.

\(\left( \begin{array}{l}
1\\
2\\
3
\end{array} \right)\)     A1     N1

[1 mark]

b.

choosing correct direction vectors     A1A1

e.g. \(\left( {\begin{array}{*{20}{c}}
3\\
{ - 2}\\
1
\end{array}} \right)\) , \(\left( {\begin{array}{*{20}{c}}
{ - 1}\\
2\\
{ - a}
\end{array}} \right)\)

recognizing that \({\boldsymbol{a}} \bullet {\boldsymbol{b}} = 0\)     M1

correct substitution     A1

e.g. \( - 3 - 4 - a = 0\)

\(a = - 7\)     A1     N3

[5 marks]

c.

Examiners report

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

Syllabus sections

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