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?

• How can vectors be used to find the midpoint of two vectors?

• 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