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

s

v

a

F

E

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 + acCommutative ab = ba

Associative (ab)c = a(bc)

Addition of scalars follows these properties:

Commutative a+b = b+aAssociative (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

AB

A+B

B+A

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

A

B

CA+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 Productor 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 Productor 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 (vxB)

where q is the charge on a proton, v is 3x105 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 10-11 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.

xx

Did you realize that you have been using a right-handed Cartesian

coordinate system in mathematics all these years?

x

y

z

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

ˆ ˆ ˆx y z

Here are a few for practice:

ˆ) ) ?

ˆ ˆ1. ?

ˆˆ2. ?

ˆ ˆ3. ?

ˆ ˆ4. 2 3 ?

ˆ5. (4 (5meters meters z

x z

z y

x y

x y

y

yx

z

ˆ6z2 ˆ20 m x

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

ˆ ˆ ?

ˆ ˆ ?

ˆ ˆ ?

ˆ ˆ ?

ˆ ˆ2 4 ?

ˆ ˆ(3 ) (4 ) ?

x x

x y

y z

z z

x x

meters x meters x

1

0

0

1

8

12 m2

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?

W F s

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.