In a science lab, a sensor is fixed at \(O\). A camera is mounted at \(A\) and a light source at \(B\), shown in the diagram. Relative to \(O\), \(\overrigh...
Question 1 Report
In a science lab, a sensor is fixed at \(O\). A camera is mounted at \(A\) and a light source at \(B\), shown in the diagram. Relative to \(O\), \(\overrightarrow{OA} = \mathbf{a}\) and \(\overrightarrow{OB} = \mathbf{b}\). \(M\) is the midpoint of \(OA\) and \(N\) is the midpoint of \(OB\).
- Find \(\overrightarrow{MN}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\). (3)
- Hence show that \(MN\) is parallel to \(AB\), and state the ratio \(MN:AB\). (2)
Answer Details
Position vectors from a fixed origin combine by vector addition and subtraction, and if one vector is a scalar multiple of another, the two corresponding line segments are parallel, with the scalar giving the ratio of their lengths.
(a) Since \(M\) is the midpoint of \(OA\), its position vector is half of \(\mathbf{a}\):
\[\overrightarrow{OM}=\dfrac{1}{2}\mathbf{a}\]Similarly, since \(N\) is the midpoint of \(OB\):
\[\overrightarrow{ON}=\dfrac{1}{2}\mathbf{b}\]The vector from \(M\) to \(N\) is found by subtracting the position vector of the start point from that of the end point:
\[\overrightarrow{MN}=\overrightarrow{ON}-\overrightarrow{OM}=\dfrac{1}{2}\mathbf{b}-\dfrac{1}{2}\mathbf{a}=\dfrac{1}{2}(\mathbf{b}-\mathbf{a})\] [3 marks](b) By the same subtraction rule, \(\overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}=\mathbf{b}-\mathbf{a}\). Comparing this with the result from part (a):
\[\overrightarrow{MN}=\dfrac{1}{2}\overrightarrow{AB}\]Since \(\overrightarrow{MN}\) is a scalar multiple of \(\overrightarrow{AB}\), the two segments are parallel. Because the scalar is \(\dfrac{1}{2}\), the ratio of their lengths is:
\[MN:AB=1:2\] [2 marks]This result is the midpoint theorem for triangles: the segment joining the midpoints of two sides of a triangle is always parallel to the third side and exactly half its length, a pattern that vector subtraction proves directly without needing any angle or coordinate measurements.
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