Vector Basics (Edexcel IGCSE Further Maths)

What is a vector?

• Vectors represent a movement of a certain distance (magnitude) in a given direction
• In one dimension, the sign of a number represents its direction 
  • For example, two objects with velocities 7 m/s and ‑7 m/s are travelling:
    • at the same speed (magnitude)
    • but in opposite directions
• In two dimensions, vectors consist of x- and y-components
  
  • This shows movement parallel to the x  and y-axes
  • Components can be positive or negative 

What is a scalar?

• A scalar is an ordinary number that does not represent movement
  • A scalar is not a vector

• Scalars can still be negative
  • For example
    • temperature is a scalar (it's -2°C today)
    • but change in temperature is a vector (it's +3°C from yesterday)

How do I write vectors?

• If you know the two components of a vector, you can write it as
  • either a column vector
    • is 3 right and 2 up
  • or in i and j notation
    • 
• If you do not know the components, use a lower-case letter to represent the entire vector
  • Exams use bold letters
    • a, b, ...
  • You should write underlined letters
    • a, b, ...
• If points (A , B , ...) are given, means the vector from A  to B
  • The order matters

How do I add vectors?

• To add vectors you add their components
  • For column vectors, add the tops together and bottoms together
    • 
• In i and j notation, add i and j parts separately
  • (2i + j) + (i + 4j) = (3i + 5j)
• Visually, the vector a + b is the shortest route
  • from the start of a
  • to the end of b

How do I subtract vectors?

• To subtract vectors you subtract their components
  
  • 
• In i and j notation, subtract i and j parts separately
  • Be careful with brackets and negatives
    • (2i + j) - (i + 4j) = (i - 3j)
• Visually, the vector a - b is the shortest route
  • from the start of a
  • to the end of -b

How do I add and subtract vectors on diagrams?

• Think of travelling along vectors as a journey
  • a + b means follow a, then follow b
    • This is the same as b + a
    • They both end up at the same place
• The diagram below shows vectors s, t and u
  • To get from A  to C  do s then t
    • s + t
• If you have to travel the wrong direction along a vector, add the negative of that vector
  •  To get from B  to D  do t then the reverse of u
    • t + (-u)
    • This simplifies to t - u
• A vector plus its negative gives the zero vector, 0
  • a + (-a) = 0
    • The zero is bold (or underlined)

How do I multiply a vector by a scalar?

• The vector a multiplied by a scalar (constant number) k  is ka
  • ka is parallel to a
  • k is the scale factor of enlargement
  • A negative k reverses the direction
• Multiply all components by the scalar constant
  • For example
    • 
    • 2(4i + 5j) = 8i + 10j