Vectors
Column vectors, magnitudes and (Higher) using vectors to construct geometric proofs.
Learn
Vectors, explained point by point
Everything the GCSE specification expects you to be able to do, and how to actually do it - the same lesson a signed-in student studies from.
Read and write column vectors, and use them to describe movements
A column vector stacks the x-move over the y-move: (3 over −2) means 3 right and 2 down. Vectors describe translations and journeys - where you end up, regardless of the route taken.
Add and subtract vectors, and multiply a vector by a number
Add or subtract matching components: (2, 5) + (3, −1) = (5, 4) - one journey followed by another. Multiplying by a scalar stretches it: 3a is three times as long as a, and −a points exactly the opposite way.
Write vectors in terms of given vectors a and b, and simplify vector expressionsHigher only
Trace a route through points you know: if AB = a and AC = b, then BC = BA + AC = −a + b. Any route between the same two points gives the same vector, so pick the one through labelled points and simplify like algebra.
Use vectors to show that lines are parallel or points lie on a straight lineHigher only
Two vectors are parallel when one is a multiple of the other: 2a + 4b is parallel to a + 2b because it is exactly twice it. To show points are collinear, show the vectors between them are parallel AND share a point - then state both facts in words.
Practice
Try a Vectors question
A GCSE-style original question from this topic. Have a go before you open the working - deciding on an answer first is what makes the working stick.
The vector (x, 6) has magnitude 10, and x is positive. Work out the value of x.
- x = 8
- x = 4
- x = 16
- x = 64
Show the answer and the working
Answer: x = 8
x² + 36 = 100, so x² = 64 and x = 8.
- Magnitude formula: = 10.
- Square both sides: x² + 36 = 100.
- Solve: x² = 64.
- x is positive, so x = = 8.