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Date May 2011 Marks available 2 Reference code 11M.1.sl.TZ2.3
Level SL only Paper 1 Time zone TZ2
Command term Find Question number 3 Adapted from N/A

Question

The following diagram shows quadrilateral ABCD, with \(\overrightarrow {{\rm{AD}}} = \overrightarrow {{\rm{BC}}} \) , \(\overrightarrow {{\rm{AB}}} = \left( \begin{array}{l}
3\\
1
\end{array} \right)\) , and \(\overrightarrow {{\rm{AC}}} = \left( \begin{array}{l}
4\\
4
\end{array} \right)\) .


Find \(\overrightarrow {{\rm{BC}}} \) .

[2]
a.

Show that \(\overrightarrow {{\rm{BD}}} = \left( {\begin{array}{*{20}{c}}
{ - 2}\\
2
\end{array}} \right)\) .

[2]
b.

Show that vectors \(\overrightarrow {{\rm{BD}}} \) and \(\overrightarrow {{\rm{AC}}} \) are perpendicular.

[3]
c.

Markscheme

evidence of appropriate approach     (M1)

e.g. \(\overrightarrow {{\rm{AC}}} - \overrightarrow {{\rm{AB}}} \) , \(\left( \begin{array}{l}
4 - 3\\
4 - 1
\end{array} \right)\)

\(\overrightarrow {{\rm{BC}}} = \left( \begin{array}{l}
1\\
3
\end{array} \right)\)    
A1     N2

[2 marks]

a.

METHOD 1

\(\overrightarrow {{\rm{AD}}} = \left( {\begin{array}{*{20}{c}}
1\\
3
\end{array}} \right)\)    
 (A1)

correct approach     A1

e.g. \(\overrightarrow {{\rm{AD}}} - \overrightarrow {{\rm{AB}}} \) , \(\left( \begin{array}{l}
1 - 3\\
3 - 1
\end{array} \right)\)

\(\overrightarrow {{\rm{BD}}} = \left( {\begin{array}{*{20}{c}}
{ - 2}\\
2
\end{array}} \right)\)    
AG     N0

METHOD 2

recognizing \(\overrightarrow {{\rm{CD}}} = \overrightarrow {{\rm{BA}}} \)     (A1)

correct approach     A1

e.g. \(\overrightarrow {{\rm{BC}}} + \overrightarrow {{\rm{CD}}} \) , \(\left( \begin{array}{l}
1 - 3\\
3 - 1
\end{array} \right)\)

\(\overrightarrow {{\rm{BD}}} = \left( {\begin{array}{*{20}{c}}
{ - 2}\\
2
\end{array}} \right)\)    
AG     N0

[2 marks]

b.

METHOD 1

evidence of scalar product     (M1)

e.g. \(\overrightarrow {{\rm{BD}}} \bullet \overrightarrow {{\rm{AC}}} \) , \(\left( {\begin{array}{*{20}{c}}
{ - 2}\\
2
\end{array}} \right) \bullet \left( \begin{array}{l}
4\\
4
\end{array} \right)\)

correct substitution     A1

e.g. \(( - 2)(4) + (2)(4)\) , \( - 8 + 8\)

\(\overrightarrow {{\rm{BD}}} \bullet \overrightarrow {{\rm{AC}}} = 0\)     A1

therefore vectors \(\overrightarrow {{\rm{BD}}} \) and \(\overrightarrow {{\rm{AC}}} \) are perpendicular     AG     N0

METHOD 2

attempt to find angle between two vectors     (M1)

e.g. \(\frac{{{\boldsymbol{a}} \bullet {\boldsymbol{b}}}}{{{\boldsymbol{ab}}}}\)

correct substitution     A1

e.g. \(\frac{{( - 2)(4) + (2)(4)}}{{\sqrt 8 \sqrt {32} }}\) , \(\cos \theta  = 0\)

\(\theta  = {90^ \circ }\)     A1

therefore vectors \(\overrightarrow {{\rm{BD}}} \) and \(\overrightarrow {{\rm{AC}}} \) are perpendicular     AG     N0

[3 marks]

c.

Examiners report

This question on two-dimensional vectors was generally very well done.

a.

This question on two-dimensional vectors was generally very well done. A very small number of candidates had trouble with the "show that" in part (b) of the question.

b.

Nearly all candidates knew to use the scalar product in part (c) to show that the vectors are perpendicular.

c.

Syllabus sections

Topic 4 - Vectors » 4.2 » Perpendicular vectors; parallel vectors.
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