IB DP Maths: AA HL复习笔记3.9.4 The Scalar Product

What is the scalar product?

• The scalar product (also known as the dot product) is one form in which two vectors can be combined together
• The result of taking the scalar product of two vectors is a real number
  • i.e. a scalar
• The scalar product of two vectors gives information about the angle between the two vectors
  • If the scalar product is positive then the angle between the two vectors is acute (less than 90°)
  • If the scalar product is negative then the angle between the two vectors is obtuse (between 90° and 180°)
  • If the scalar product is zero then the angle between the two vectors is 90° (the two vectors are perpendicular)

How is the scalar product calculated?

• There are two methods for calculating the scalar product
• The most common method used to find the scalar product between the two vectors v and w is to find the sum of the product of each component in the two vectors

What properties of the scalar product do I need to know?

How do I find the angle between two vectors?

• If two vectors with different directions are placed at the same starting position, they will form an angle between them
• The two formulae for the scalar product can be used together to find this angle
  • This is given in the formula booklet
• To find the angle between two vectors:
  • Calculate the scalar product between them
  • Calculate the magnitude of each vector
  • Use the formula to find cos θ
  • Use inverse trig to find θ

How do I know if two vectors are perpendicular?

• If the scalar product of two (non-zero) vectors is zero then they are perpendicular
• Two vectors are perpendicular if their scalar product is zero
  • The value of cos θ= 0 therefore |v||w|cos θ= 0