3.7.5 The Vector Product

What is the vector (cross) product?

• The vector product (also known as the cross product) is a form in which two vectors can be combined together
• The vector product between two vectors v and w is denoted v × w
• The result of taking the vector product of two vectors is a vector
• The vector product is a vector in a plane that is perpendicular to the two vectors from which it was calculated
• This could be in either direction, depending on the angle between the two vectors
• The right-hand rule helps you see which direction the vector product goes in
• By pointing your index finger and your middle finger in the direction of the two vectors your thumb will automatically go in the direction of the vector product

How do I find the vector (cross) product?

• There are two methods for calculating the vector product
• The vector product of the two vectors v and w can be written in component form as follows:
  •  
  • Where  and 
  • This is given in the formula booklet
• The vector product can also be found in terms of its magnitude and direction
• The magnitude of the vector product is equal to the product of the magnitudes of the two vectors and the sine 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
  • This is given in the formula booklet
• The direction of the vector product is perpendicular to both v and w

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

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

How do we find the shortest distance from a point to a line?

• The vector product can be used to find the shortest distance from any point to a line on a 2-dimensional plane
• Given a point, P, and a line r = a + λb
• The shortest distance from P to the line will be
• 
• Where A is a point on the line

How do I use the vector product to find the area of a parallelogram?

• The area of the parallelogram with two adjacent sides formed by the vectors v and w is equal to the magnitude of the vector product of two vectors v and w
• where v and w form two adjacent sides of the parallelogram
• This is given in the formula booklet

How do I use the vector product to find the area of a triangle?

• The area of the triangle with two sides formed by the vectors v and w is equal to half of the magnitude of the vector product of two vectors v and w
• where v and w form two sides of the triangle
• This is not given in the formula booklet