Add/subtract column vectors componentwise
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 , written with modulus signs as — 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.
A reported slip: "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.
A reported error: "Some students reversed the x and y elements for one of the vectors before adding or subtracting them."
Not knowing where to start on a 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.
A reported difficulty: "Many students found this difficult, often not knowing where to start. Several students added the two given vectors and then stopped."
Magnitude of a sum is not additive
In general because vectors have direction. Add the components first to get one column vector, then take . The two are equal only when and point the same way.
Reading k from only one row
If , both rows must give the same : check it from the top and the bottom. If they disagree, is not a scalar multiple of . Reading off one row only can hand you a false value.
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.
Add legs first, then take the magnitude
For a straight-line distance across several legs (e.g. ), add the legs into one column vector first, then apply Pythagoras. The magnitude is not the sum of the leg lengths.
p·a + q·b = c: solve simultaneously
To find scalars and with , write the left side as one column vector, equate the top rows and the bottom rows to , and solve the two simultaneous equations.
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.
