Cambridge O Level Additional Mathematics · Syllabus 4037 · Vectors in Two Dimensions
Position Vector
What is Position Vector?
The vector from a fixed origin O to a point, written OA with an arrow above it or as the bold letter a. Its components are the coordinates of the point, so the point A with coordinates (6, 1) has position vector (6, 1) measured from O. A position vector fixes where a point is; a displacement vector fixes how to travel from one point to another and is found by subtracting the start point's position vector from the endpoint's, giving AB = b minus a. Moving the origin changes every position vector but leaves every displacement vector unchanged.
This definition is part of the Vectors in Two Dimensions chapter in Cambridge O Level Additional Mathematics.
Common mistakes with Position Vector
- Writing \(\mathbf r_A=\mathbf r_{A0}+t\mathbf v_A\) and \(\mathbf r_B=\mathbf r_{B0}+s\mathbf v_B\) when testing for a collision. Why it fails Two independent letters permit the particles to arrive at different moments, so the equations can be satisfied by a near miss. The set-up has quietly answered “do the paths cross?” instead, and it will usually return a solution — which is precisely the trap. Fix One clock for both particles. Use the same \(t\) in both position vectors from the first line, and the requirement of a common time is built into the algebra rather than remembered at the end.
- Answering “yes, at \(t=2\)” when the question asked when and where they collide. Why it fails The position is a separate piece of information, and a question that asked for both has only been half answered. It also serves as the check on the time: substituting \(t\) into both position vectors and getting the same point confirms the whole calculation. Fix Finish every collision answer by substituting the common \(t\) into both position vectors and stating the point. Also check that \(t\) is admissible — a negative \(t\) means the meeting would have happened before the motion started, so the particles do not collide.
Questions students ask about Position Vector
How do you find the displacement vector \(\overrightarrow{AB}\) from two position vectors?
Subtract the start point's position vector from the endpoint's: \(\overrightarrow{AB}=\mathbf b-\mathbf a\), which comes from the route \(\overrightarrow{AO}+\overrightarrow{OB}=-\mathbf a+\mathbf b\). Get the order backwards and every component's sign flips, reversing every direction, angle and unit vector calculated from it afterwards. \(\overrightarrow{BA}=-\overrightarrow{AB}\) follows automatically: it is the same line but not the same vector, since the two point in opposite directions.
How do you decide whether two moving particles actually collide?
Write each particle's position as \(\mathbf r=\mathbf r_0+t\mathbf v\), using the same variable \(t\) for both, then set the two position vectors equal to get one equation from the \(x\)-components and one from the \(y\)-components. A collision needs both equations to give the same value of \(t\); if they give different values, the paths cross geometrically but the particles pass through that point at different moments, which is a miss, not a collision.