Add/subtract vectors component-wise
For add (or subtract) the tops and the bottoms separately. Scalar multiple multiplies BOTH components.
Magnitude is Pythagoras on the components
The length (magnitude) of is — the squares make signs irrelevant, so a negative component does not give a negative length.
Drawn from real examiner reports.
Finding the vector but then stopping
When asked for a length, students often work out the correct column vector (e.g. for ) and write that on the answer line, forgetting to go on and apply Pythagoras to get the magnitude.
Seen on Nov 2024 Paper 2H (Q17): "Some gained the correct vector for HF and gained 1 mark, but could go no further. A small number correctly used Pythagoras and gained the correct answer of 15."
Reversing the x and y components
Swapping the top and bottom of one vector before adding or subtracting scrambles the result.
Reported on Nov 2024 Paper 2H (Q17): "Some students reversed the x and y elements for one of the vectors before adding or subtracting them."
Not knowing where to start with a vector path
Many simply added the two given vectors and assumed that was the answer, rather than building the required vector by following a route through the labelled points.
Reported on Nov 2024 Paper 2H (Q17): "Many students found this difficult, often not knowing where to start. Several students added the two given vectors and then stopped."
Magnitude cannot be negative
A magnitude is a length, so it is never negative. Squaring each component makes it positive first, so a component such as −8 contributes , not a minus. Never carry a minus sign out of the square root.
Subtracting a − b the wrong way round
For , subtract the components of from those of : top − top and bottom − bottom. Doing instead gives the negative of the correct vector, flipping every sign.
Read off the route, then compute
Write the required vector as a sum of journeys you know (e.g. ), substitute the columns, simplify, and only then take the magnitude if a length is asked for.
Square components before adding
For a magnitude, square each component first (this removes any minus sign), add the squares, then take the square root. Skipping the squaring, or rooting too early, is where length errors creep in.
Vector or length — answer what is asked
Check whether the question wants a vector or a length. Give the column vector when a vector is asked; go on and apply Pythagoras for the magnitude when a length is asked. Do not stop halfway.
A vector means " across, up" (down/left are negative). Bold or both name vectors.
How do you add two column vectors?
Given and , work out as a column vector.
