The?scalar?product?is an important link between the algebra of vectors and the trigonometry of vectors. We shall see that the scalar product is somewhat comparable to the operation of multiplication on real numbers.

What is the scalar (dot) product?

• The scalar product between two vectors?a?and?b?is represented by
  • This is also called the dot product because of the symbol used
• The scalar product between two vectors ?and ?is defined as
• The result of taking the scalar product of two vectors is a?real number
  • i.e. a?scalar
• For example,

• The scalar product has some important properties:
  • The order of the vectors doesn’t affect the result:

• In effect we can ‘multiply out’ brackets:

• This means that we can do many of the same things with vectors as we can do when operating on real numbers – for example,

• The scalar product between a vector and itself is equal to the square of its?magnitude:

For example,

What is the connection between the scalar product and trigonometry?

• There is another important method for finding involving the angle between the two vectors θ:

• • Here θ?is the angle between the vectors when they are placed?‘base to base’
    • when the vectors are placed so that they begin at the same point
  • This formula can be derived using the cosine rule and expanding
• 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 do I tell if vectors or lines are perpendicular?

• Two (non-zero) vectors a and b?are?perpendicular if, and only if,
  • If the?a?and?b?are perpendicular then:
    • 
  • If? ?then:
    • ?are perpendicular
  • For example, the vectors ??are perpendicular since

How do I find the angle between two vectors?

• Recall that a formula for the scalar (or ‘dot’) between vectors a and b?is

• • where θ?is the angle between the vectors when they are placed?‘base to base’
    • that is, when the vectors are positioned so that they start at the same point
  • We arrange this formula to make ?the subject:
  • To find the angle between two vectors
    • Calculate the scalar product between them
    • Calculate the magnitude of each vector
    • Use the formula to find
    • Use inverse trig to find θ

How do I find the angle between two lines?

• To find the angle between two lines, find the angle between their?direction vectors

• • ?For example, if the lines have equations ?and , then the angle θ?between the lines is given by