The Scalar Product (DP IB Maths: AI HL)

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 scalar product between two vectors a and b is denoted 
• 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
• 
• Where and 
• This is given in the formula booklet
• The scalar product is also equal to the product of the magnitudes of the two vectors and the cosine of the angle between them
• 
• Where θ is the angle between v and w
• The two vectors v and w are joined at the start and pointing away from each other
• The scalar product can be used in the second formula to find the angle between the two vectors

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

• If two vectors, v and w, are parallel then the magnitude of the scalar product is equal to the product of the magnitudes of the vectors
  • 
  • This is because cos 0° = 1 and cos 180° = -1
• If two vectors are perpendicular the scalar product is zero
  • This is because cos 90° = 0

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
• If  then v and w must be perpendicular to each other
• Two vectors are perpendicular if their scalar product is zero
• The value of cos θ = 0 therefore |v||w|cos θ = 0