Vectors (Edexcel GCSE Maths): Revision Notes
Vectors
What are vectors?
A vector is a mathematical quantity that has two important properties:
- Magnitude (also called size or length)
- Direction
Vectors are different from ordinary numbers because they tell us both how much of something there is and which way it's pointing.
Think of vectors like arrows - the length of the arrow shows the magnitude, and the way it points shows the direction. This visual representation helps make the abstract concept more concrete.
Vector notation
Vectors can be written in several ways:
- Bold letters: a, b, c
- Letters with arrows:
- Column vectors:
When you see a vector written as , this means the vector goes from point A to point B.
Scalar multiplication of vectors
You can multiply a vector by a number (called a scalar). When you do this:
- The direction stays the same
- The magnitude changes
Worked Example: Scalar Multiplication
If vector a has a certain length, then:
- 3a has three times the length but points in the same direction
- 0.5a has half the length but points in the same direction
- -2a has twice the length but points in the opposite direction
Negative vectors
If b is a vector, then -b is a vector that:
- Has the same magnitude as b
- Points in the opposite direction to b
This is like turning the arrow around completely - imagine flipping an arrow 180 degrees while keeping its length exactly the same.
Adding vectors using the triangle law
The triangle law is a method for adding vectors together. This method works because you're essentially following a path - you travel along the first vector, then along the second vector, and the resultant shows your overall displacement.
Worked Example: Triangle Law
Here's how the triangle law works:
Step 1: Draw the first vector
Step 2: From the end of the first vector, draw the second vector
Step 3: The resultant vector goes from the start of the first vector to the end of the second vector
The equation is: a + b = c, where c is the resultant vector.
Column vectors
Column vectors represent vectors using coordinates. They show how far the vector moves horizontally and vertically.
A column vector means:
- Move x units horizontally (right if positive, left if negative)
- Move y units vertically (up if positive, down if negative)
Operations with column vectors
Worked Example: Column Vector Operations
Multiplying by a scalar:
You multiply both parts by the scalar.
Adding column vectors:
You add the top numbers together and add the bottom numbers together.
Working with vector problems
When solving vector problems, these strategies will help you work systematically:
Problem-Solving Strategies:
- Trace paths carefully - follow the direction of each vector
- Use parallel vectors - if two sides of a shape are parallel, their vectors are equal (like )
- Opposite directions mean subtraction - if you go against a vector's direction, you subtract it
- Simplify your answers - combine like terms where possible
Key relationships in shapes
In parallelograms and other shapes:
- Opposite sides have equal vectors
- To go from one point to another, you can follow different paths that give the same result
- When vectors form a closed loop, they add up to zero
Common Mistake to Avoid: Always pay attention to the direction of vectors when following paths. Going against a vector's direction means you must subtract it, not add it.
Exam tips
Essential Exam Strategies:
- Always show your working clearly
- Draw diagrams when they help you visualise the problem
- Check that your final vector makes sense by looking at the direction and magnitude
- Remember that vector addition is commutative: a + b = b + a
Key Points to Remember:
- Vectors have both magnitude and direction - this makes them different from ordinary numbers
- Use the triangle law to add vectors by placing them end-to-end
- Column vectors make calculations easier - just add or multiply the corresponding parts
- Negative vectors point in the opposite direction with the same length
- Always trace your path carefully when working through vector problems