Vector Addition (Cambridge International IGCSE Further Maths)

Vector addition

What is vector addition?

• Adding vectors together lets us describes the movement between two points

• To add or subtract vectors numerically simply add or subtract each of the corresponding components

• In column vector notation just add the top, middle and bottom parts together

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

• In base vector notation add each of the i and j components together separately

  • For example: (2i + j) – (i + 4j) = (i – 3j)

• Adding vectors creates a single vector which is called the resultant vector

  • The resultant vector will be the shortest route from the start of the first vector to the end of the second

• Subtracting a vector is the same as adding a negative vector

• Adding the vectors PQ and QP gives the zero vector, denoted by a bold zero 0(0 in handwriting)

What are scalars and parallel vectors?

• Two vectors are parallel if and only if one is a scalar multiple of the other

  • i.e both components of the vector have been multiplied by the same constant

• Multiplying a vector by a positive scalar changes the magnitude (size) but not its direction

• Multiplying a vector by a negative scalar changes the magnitude and the direction would be reversed

How do I find the vector between two points?

• If, relative to the origin $O$, the points $A$ and $B$ have position vectors

  • $\overset{\rightarrow}{OA}=a$
  • $\overset{\rightarrow}{OB}=b$

then

• $\overset{\rightarrow}{AB}=\overset{\rightarrow}{AO}+\overset{\rightarrow}{OB}=-\overset{\rightarrow}{OA}+\overset{\rightarrow}{OB}=-a+b=b-a$

• Similarly, $\overset{\rightarrow}{BA}=\overset{\rightarrow}{BO}+\overset{\rightarrow}{OA}=a-b$

• This result is particularly useful when working with position vectors (as the 'journey' can always go via the origin)

  • but the result applies to any set of three vectors

Exam Hint

• Think of vectors like a journey from one place to another

  • You may have to take a detour eg. A to B might be A to O then O to B.

• Diagrams can help, so if there isn’t one, draw one

  • If there are any, labelling parallel vectors will help

Worked Example