Pearson Edexcel IAL Mathematics P4.7.4 Position vectors

Practise using position vectors to move between points, find AB from OB - OA, and recover coordinates from vector sums in 2D and 3D.

Exam points

calculate AB by subtracting position vectors in i, j, k or column form
find a missing point by adding a displacement vector to a known position vector
use a line parameter to show or find possible position vectors for a point

[Maximum number: 3]

Relative to a fixed origin O, the points A, B and C have position vector a, b and c respectively.

Points A, B and C lie in a straight line, with B lying between A and C.
Given A B: A C=1: 3 show that

\mathbf{c}=3 \mathbf{b}-2 \mathbf{a}

M1: Attempts any two of $\overrightarrow{A B}, \overrightarrow{A C}$ and $\overrightarrow{B C}$ .
Condone the wrong way around but it must be subtraction.
Allow marked in the correct place on a diagram
dM1: Uses the given information.
Accept $\overrightarrow{A B}=\frac{1}{3} \overrightarrow{A C}, \quad \overrightarrow{B C}=2 \times \overrightarrow{A B}, \quad \overrightarrow{B C}=\frac{2}{3} \times \overrightarrow{A C} \quad$ etc condoning slips as in previous M1.
A1*: Fully correct work inc bracketing leading to the given answer c=3 b-2 a
Expect to see the brackets multiplied out. So $\mathbf{c}-\mathbf{b}=2 \times(\mathbf{b}-\mathbf{a}) \Rightarrow \mathbf{c}-\mathbf{b}=2 \mathbf{b}-2 \mathbf{a} \Rightarrow \mathbf{c}=3 \mathbf{b}-2 \mathbf{a}$ is fine.

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P4.7.4 - Position vectors question 1