Photo AI
The diagram shows triangle OAB and points C and D - OCR - GCSE Maths - Question 16 - 2019 - Paper 4
Question 16
The diagram shows triangle OAB and points C and D. OA = 3a and OB = 3b. C lies on AB such that AC = 2CB. D is such that BD = 2a + b. Show, using vectors, that OCD ... show full transcript
Worked Solution & Example Answer:The diagram shows triangle OAB and points C and D - OCR - GCSE Maths - Question 16 - 2019 - Paper 4
Step 1
AC = 2CB
Answer
Let the position vector of point C be represented as vector OC. Then, we can express AC in terms of the position vectors:
-
From the triangle, we have:
- Vector AC = Vector C - Vector A
- Vector C = Vector OC = OA + CB
-
Given that AC = 2CB, we can set up the equation:
-
We can express Vector CB as:
- Vector CB = Vector B - Vector C = OB - OC, where OB = 3b.
-
Thus, we have: and substituting in:
Step 2
BD = 2a + b
Answer
-
The position vector for point D can be expressed as:
-
Since BD is defined as the difference in position vectors, we can write:
-
Now, we need to show that points O, C, and D are collinear. For this, we can express vectors OC and OD in terms of scalars: for some scalar k.
-
Thus, by showing that the ratios of the components between OC and OD are equal, we can conclude that OCD forms a straight line.