Vector Basics (Edexcel IGCSE Further Maths)

Vector Notation

Spec point — 5A

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

    • $( \begin{gathered} 3 \\ 2 \end{gathered} )$ is 3 right and 2 up

  • or in i and j notation

    • $3i+2j$

• 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, $\overset{\rightarrow}{AB}$ means the vector from A to B

  • The order matters

Exam Hint

• Diagrams can help. If there isn’t one, try sketching one!

Adding & Subtracting Vectors

Spec point — 7A

How do I add vectors?

• To add vectors you add their components

  • For column vectors, add the tops together and bottoms together

    • $( \frac{2}{1} ) + ( 1 \frac{1}{4} ) = ( \frac{3}{5} )$

• 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

  • $( \frac{2}{1} ) - ( 1 \frac{1}{4} ) = ( \frac{1}{- 3} )$

• 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

    • $2 ( \begin{gathered} 4 \\ 5 \end{gathered} ) = ( \begin{gathered} 8 \\ 10 \end{gathered} )$
    • 2(4i + 5j) = 8i + 10j

Exam Hint

• In the exam, questions with lots of parts usually build on each other.

  • Check if any previous results give shortcuts!

Worked Example Two vectors are given by a = 4i + 2j and b = *pi – **j **where p is an unknown constant.

If a + 2b = –6i, find the value of p.

Substitute the vectors into the left-hand side

*(4i + 2j) + 2(pi *– j)

Expand the brackets

4i + 2j + 2pi* – 2j

Add the vectors by adding the components The 2j and -2j cancel

*(4 + 2p)i

This must equal the right-hand side of –6**i Write down an equation in p to balance both sides

*4 + 2**p *= –6

*Solve it to find *p

2p = –10

Final answer: p = –5