Vectors and Vector Multiplication
Dec 31, 2015
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)
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
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
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.