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Problem Solving using Vectors (Cambridge O Level Additional Maths)
Revision Note
Problem Solving using Vectors
What problems may I be asked to solve involving vectors?
- Showing that two lines or vectors are parallel
- Two vectors are parallel if they are scalar multiples of each other
- i.e. where is a constant
- See Vector Addition
- Finding the midpoint of two (position) vectors
- Showing that three points are collinear
- Collinear describes points that lie on the same straight line
- e.g. The points and all lie on the line with equation
- Vectors can be used to show this, and similar, results
- Collinear describes points that lie on the same straight line
- Results concerned with geometric shapes
- Shapes with parallel lines are often involved
- e.g. parallelogram, rhombus
- These often include lines or vectors being split into ratios
- e.g. The point lies on the line such that
- Shapes with parallel lines are often involved
How do I find the midpoint of two vectors?
- If the point has position vector and the point has position vector
- the position vector of the midpoint of is
- This can be derived by considering
-
- using the result from Vector Addition
- If is the midpoint of then
- Therefore, the position vector of the midpoint, is
-
How do I show three points are collinear?
- Three points are collinear if they all lie on the same straight line
- There are two ways to show this for three points, and say
- Method 1
Show that where is a constant
i.e. show that and are scalar multiples of each other- As the vectors are scalar multiples they will have the same direction (and so be parallel)
- So as both vectors start at point , they must be collinear
- Method 2
Show that AND that point lies on both the vectors and
- Method 1
- Which method you should use will depend on the information given and how you happen to see the question
How do I solve problems involving geometric shapes?
- Problems involving geometric shapes involve finding paths around the shape using known vectors
- there will be many other vectors in the shape that are equal and/or parallel to the known vectors
- The following grid is made up entirely of parallelograms, with the vectors and defined as marked in the diagram:
- Note the difference between "specific" and "general" vectors
- The vector in the diagram is specific and refers only to the vector starting at and ending at
- However, the vector is a general vector
- any vector the same length as and parallel to it is equal to
- e.g.
- Vector is also a general vector
- e.g.
- However, the vector is a general vector
- There will also be vectors in the diagram that are the same magnitude but have the opposite direction to or
- e.g.
- The vector in the diagram is specific and refers only to the vector starting at and ending at
- There are also many instances of the vector addition result
- e.g.
- There are many scalar multiples of the vectors or
- e.g.
- Using a combination of these it is possible to describe a vector between any two points in terms of and
Examiner Tip
- Diagrams are helpful in vector questions
- If a diagram has been given, label it and add to it as you progress through a question
- If a diagram has not been given, draw one, it does not need to be accurate!
Worked example
The following diagram consists of a grid of identical parallelograms.
Vectors and are defined by and .
Write the following vectors in terms of and .
a)
To get from A to E follow vector four times (to the right).
b)
There are many ways to get from G to T.
One option is to go from G to Q ( twice), and then from Q to T ( three times).
One option is to go from G to Q ( twice), and then from Q to T ( three times).
c)
Point is such that it is midpoint of .
Find the vector .
Find the vector .
There are many ways to get from P to Z.
One option is to go from P to R ( twice), and then from R to Z ( one-and-a-half times).
One option is to go from P to R ( twice), and then from R to Z ( one-and-a-half times).
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