Shape and Space 8
4 marks · NormalIn the diagram below, OABC is a parallelogram. , . D is a point such that .
Express in terms of and .
Find in terms of and .
Show that points O, D and C are collinear.
Work this out on paper, then compare with the answer and worked solution below. With a free account you can submit written answers for marking.
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(a)
(b)
(c) O, D and C are collinear because ; C lies on line OD. · must show states or shows , compares and , concludes collinearity
Common mistakes: Mixing up vector directions; Missing negative signs; Not justifying collinearity
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Marking points
B1 (a) · 1 mark · Correctly gives
B2 (b) · 1 mark · Correctly gives
M1 (c) · 1 mark · Finds in terms of and and links it to
A1 (c) · 1 mark · Clearly concludes points are collinear with justification
Worked solution
a)
b)
c) From the parallelogram, . But for D, and for C, with the structure, we can check if C lies on OD.
Writing in terms of and using the parallelogram structure shows that C, O and D are collinear if is a multiple of (or vice versa)—if the expression matches, they are on the same line.