Geometric Proof with Vectors
How can vectors be used to prove geometrical properties?
- If two vectors can be shown to be parallel then this can be used to prove parallel lines
- If two vectors are scalar multiples of each other then they are parallel
- To prove that two vectors are parallel simply show that one is a scalar multiple of the other
- If two vectors can be shown to be perpendicular then this can be used to prove perpendicular lines
- If the scalar product is zero then the two vectors are perpendicular
- If two vectors can be shown to have equal magnitude then this can be used to prove two lines are the same length
- To prove a 2D shape is a parallelogram vectors can be used to
- Show that there are two pairs of parallel sides
- Show that the opposite sides are of equal length
- The vectors opposite each other with be equal
- If the angle between two of the vectors is shown to be 90° then the parallelogram is a rectangle
- To prove a 2D shape is a rhombus vectors can be used to
- Show that there are two pairs of parallel sides
- The vectors opposite each other with be equal
- Show that all four sides are of equal length
- If the angle between two of the vectors is shown to be 90° then the rhombus is a square
How are vectors used to follow paths through a diagram?
- In a geometric diagram the vector forms a path from the point A to the point B
- This is specific to the path AB
- If the vector is labelled a then any other vector with the same magnitude and direction as a could also be labelled a
- The vector would be labelled -a
- It is parallel to a but pointing in the opposite direction
- If the point M is exactly halfway between A and B it is called the midpoint of A and the vector could be labelled
- If there is a point X on the line AB such that then X is two-thirds of the way along the line
- Other ratios can be found in similar ways
- A diagram often helps to visualise this
- If a point X divides a line segment AB into the ratio p : q then
How can vectors be used to find the midpoint of two vectors?
- If the point A has position vector a and the point B has position vector b then the position vector of the midpoint of is
- The displacement vector
- Let M be the midpoint of then
- The position vector
How can vectors be used to prove that three points are collinear?
- Three points are collinear if they all lie on the same line
- The vectors between the three points will be scalar multiples of each other
- The points A, B and C are collinear if
- If the points A, B and M are collinear and then M is the midpoint of
Exam Tip
- Think of vectors like a journey from one place to another
- You may have to take a detour e.g. A to B might be A to O then O to B
- Diagrams can help, if there isn’t one, draw one
- If a diagram has been given begin by labelling all known quantities and vectors
Worked Example
Use vectors to prove that the points A, B, C and D with position vectors a = (3i – 5j – 4k), b = (8i - 7j - 5k), c = (3i - 2j + 4k) and d = (5k – 2i) are the vertices of a parallelogram.