In column form you add or subtract the top numbers together and the bottom numbers together, keeping them strictly separate. Geometrically, a + b means travelling along a then along b, tip to tail, and the resultant is the direct route from start to finish. Subtraction is addition of the reverse: a - b means going along a then backwards along b. Order matters for subtraction, so a - b and b - a point in opposite directions.
The column arithmetic is easy and almost nobody gets it wrong. What goes wrong is the diagram work: deciding which way round to subtract, and reading a route through a shape. A student who can compute a + b in two seconds can still stall on "express BC in terms of a and b".
The fix is to stop thinking of vectors as pairs of numbers and start thinking of them as journeys. Every vector question in the exam is a question about routes.
Column addition
add tops, add bottoms, separately
Column subtraction
subtract tops, subtract bottoms
Tip to tail
a + b: travel a, then travel b
Subtraction
a - b = a + (-b), reverse b
Reversing
-b has the same length, opposite direction
Route rule
AB = -BA always
The route rule is the one to hold onto. Going from A to B and going from B to A cover the same ground in opposite directions, so one is the negative of the other. Almost every diagram question uses this at least once.
Given a as the column vector (3, -1) and b as (-5, 4), find a + b, a - b and b - a.
Solution
a + b: tops give 3 + (-5) = -2, bottoms give -1 + 4 = 3. So a + b = (-2, 3).a - b: tops give 3 - (-5) = 8, bottoms give -1 - 4 = -5. So a - b = (8, -5).b - a: tops give -5 - 3 = -8, bottoms give 4 - (-1) = 5. So b - a = (-8, 5).b - a is exactly the negative of a - b, as expected.The step students get wrong
Losing the double negative. Subtracting -5 means 3 - (-5) = 3 + 5 = 8, not -2. This single slip accounts for more lost vector marks than anything else in the topic. Write the bracket in before you simplify rather than doing it in your head.
In a diagram, OA = a and OB = b, where O is the origin. Express AB in terms of a and b.
Solution
O.AB = AO + OB.AO is the reverse of OA, so AO = -a.OB = b.AB = -a + b, usually written AB = b - a.The rule worth memorising: AB = b - a, meaning the vector from A to B is always the endpoint minus the starting point. It is the destination minus the origin, in that order. Getting this backwards flips the direction and costs the mark even when the arithmetic is perfect.
To add a + b geometrically, draw a, then start b where a finished. The resultant is the arrow from the very start to the very end. This is why it is called the tip-to-tail rule.
If you draw both a and b starting from the same point instead, they form two sides of a parallelogram, and a + b is the diagonal from that shared corner. Both pictures give the same answer, so use whichever the diagram in the question suggests.
OACB is a parallelogram with OA = a and OB = b. Express OC and AB in terms of a and b.
Solution
AC = OB = b.OC = OA + AC = a + b.AB, use the endpoint-minus-start rule: AB = b - a.a + b and b - a.Does the order matter when adding vectors?
No for addition: a + b and b + a give the same resultant, because you end up at the same place either way. It matters very much for subtraction, where a - b and b - a point in opposite directions.
How do I remember AB = b - a?
Destination minus starting point. The letters in AB read A then B, but the formula uses them in reverse order, which is exactly why it is worth memorising as a phrase rather than reconstructing it under pressure.
What is a resultant?
The single vector that has the same effect as the ones you combined. If you travel along a and then along b, the resultant is the direct route that would have taken you to the same finishing point.
Can I add a vector and a number?
No. A vector has both size and direction, so adding a plain number is not defined. You can multiply a vector by a number, which stretches or reverses it, and that is a scalar multiple.
How does this connect to the rest of E-Maths?
Once you can build routes, you can find the length of the result using column vectors and magnitude. Expressing the same journey two different ways is the whole basis of proving three points are collinear.
— Mr Gan Math Tuition
Mr. Gan works through the journeys on paper until the tip-to-tail rule becomes automatic.
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