IB DP Maths: AI HL复习笔记3.7.7 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?

• 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