Vectors and Vector Multiplication

Vector quantities are those that have magnitude and direction, such as: Displacement,  x or Velocity, Acceleration, Force, Torque, Electric field, ….to name just a few

Scalar quantities have only magnitude: Speed, v Distance, d Time, t Energy, E Power, P Charge, q Electric potential, V

Multiplication of scalar quantities follows all the “usual” rules, including: Distributive a(b+c) = ab + ac Commutative ab = ba Associative (ab)c = a(bc)

Addition of scalars follows these properties: Commutative a+b = b+a Associative (a+b)+c = a+(b+c) Subtraction a+(-b) = a-b

Addition of vectors is commutative and associative and follows the subtraction rule: A+B = B+A (A+B)+C = A+(B+C) A-B = A+(-B)

A+B = B+A A B A+BA+B B+A

(A+B)+C = A+(B+C) A B C A+BA+B (A+B)+C B+C A+(B+C)

A-B = A+(-B) A B-B A-B

Multiplication of a scalar and a vector follows previous rules: aB = Ba a(B+C) = aB + aC

However, multiplication of vectors has a new set of rules—the vector cross product (or “vector product”) and the vector dot product or “scalar product”.

Vector Dot Product or Scalar Product A·B = AB cos  Essentially, this means multiplying the first vector times the component of the second vector that is in the same direction as the first vector—yielding a product that is a scalar quantity.

A B  B sin  B cos  A·B = AB cos  Multiple the magnitude of vector A times the magnitude of vector B times the cosine of the angle between them—or multiply the components that are in the same direction. The answer is a scalar with the units appropriate to the product AB.

Vector Cross Product or Vector Product AxB = AB sin  Essentially, this means multiplying the first vector times the component of the second vector that is perpendicular to the first vector—yielding a product that is a vector quantity. The direction of the new vector is found using the right hand rule.

A B  B sin  B cos  Multiple the magnitude of vector A times the magnitude of vector B times the sine of the angle between them—or multiply the components that are perpendicular. The answer is a vector with the units appropriate to the product AB and direction found by using the right hand rule.

For example, let’s take the vector cross product: F = q (v x B) where q is the charge on a proton, v is 3x10 5 m/s to the left on the paper, and B is 500 N/C outward from the paper toward you. The equation for this is also: F = qvB sin 

The answer for the force is 2.4 x newtons toward the top of the paper.

Unit vectors Unit vectors have a size of “1” but also have a direction that gives meaning to a vector. We use the “hat” symbol above a unit vector to indicate that it is a unit vector. For example, is a vector that is 1 unit in the x- direction. The quantity 6 meters is a vector 6 meters long in the x-direction.

Did you realize that you have been using a right-handed Cartesian coordinate system in mathematics all these years?

You can check your use of the right hand rule, because

Here are a few for practice:

We can also do dot products with unit vectors. Try these: m 2

The calculation of work is a scalar product or dot product: What is the work done by a force of 6 newtons east on an object that is displaced 2 meters east? What is the work done by a force of 6 newtons east on an object that is displaced 2 meters north? What is the work done by a force of 6 newtons east on an object that is displaced 2 meters at 30 degrees north of east? 12 joules zero 10.4 joules

In summary: In an equation or operation with a scalar or dot product, the answer is a scalar quantity that is the product of two vectors. The dot product is found by multiplying the components of vectors that are in the same direction. In an equation or operation with a vector or cross product, the answer is a vector quantity that is the product of two vectors. The cross product is found by multiplying the components of vectors that are perpendicular to each other.