In a laboratory, four sensor mounting points A, B, C and D have position vectors from a fixed origin O given by \(\overrightarrow{OA}=2\mathbf{a}\), \(\over...
Question 1 Report
In a laboratory, four sensor mounting points A, B, C and D have position vectors from a fixed origin O given by \(\overrightarrow{OA}=2\mathbf{a}\), \(\overrightarrow{OB}=2\mathbf{a}+3\mathbf{b}\), \(\overrightarrow{OC}=5\mathbf{a}+5\mathbf{b}\) and \(\overrightarrow{OD}=5\mathbf{a}+2\mathbf{b}\).
- Find \(\overrightarrow{AB}\) in terms of a and b. (2)
- Find \(\overrightarrow{DC}\) in terms of a and b. (2)
- Hence show that ABCD is a parallelogram. (1)
- Find the position vector of the midpoint of AC, in terms of a and b. (1)
Answer Details
Every side of quadrilateral ABCD can be written in terms of \(\mathbf{a}\) and \(\mathbf{b}\) by subtracting position vectors; showing that opposite sides \(\overrightarrow{AB}\) and \(\overrightarrow{DC}\) are identical vectors proves ABCD is a parallelogram, since equal vectors mean equal length and the same direction.
(a) \(\overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}=(2\mathbf{a}+3\mathbf{b})-2\mathbf{a}=3\mathbf{b}\). [2 marks]
(b) \(\overrightarrow{DC}=\overrightarrow{OC}-\overrightarrow{OD}=(5\mathbf{a}+5\mathbf{b})-(5\mathbf{a}+2\mathbf{b})=3\mathbf{b}\). [2 marks]
(c) Since \(\overrightarrow{AB}=\overrightarrow{DC}=3\mathbf{b}\), side AB is parallel to side DC and equal in length to it, which is exactly the condition needed for ABCD to be a parallelogram. [1 mark]
(d) The midpoint of AC has position vector \(\tfrac12(\overrightarrow{OA}+\overrightarrow{OC})=\tfrac12(2\mathbf{a}+5\mathbf{a}+5\mathbf{b})=\tfrac12(7\mathbf{a}+5\mathbf{b})=3.5\mathbf{a}+2.5\mathbf{b}\). [1 mark]
A quadrilateral is proved to be a parallelogram from vectors by finding just one pair of opposite sides and showing they are equal as vectors, not merely equal in length; being equal vectors automatically guarantees they are also parallel.
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