Transcript Vectors and Vector Multiplication

Vectors and Vector Multiplication
Vector quantities are those that have
magnitude and direction, such as:
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Displacement, x or s
Velocity, v
Acceleration, a
Force, F
Torque, 
Electric field, E
….to name just a few
Scalar quantities have only
magnitude:
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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
B
A
A+B
B+A
(A+B)+C = A+(B+C)
B
A
C
A+B
B+C
(A+B)+C
A+(B+C)
A-B = A+(-B)
B
A
A-B
-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.
B
B sin
A

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.
B
B sin
A

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, x̂ is a vector that is 1 unit in the xdirection. The quantity 6 meters x̂ 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?
ŷ
x̂
ẑ
You can check your use of the right
hand rule, because
xˆ  yˆ  zˆ
Here are a few for practice:
1. xˆ  zˆ  ?
2. zˆ  yˆ  ?
3.  xˆ  yˆ  ?
 ŷ
 x̂
 ẑ
4. 2 xˆ  3 yˆ  ? 6zˆ
5. (4meters ) yˆ  (5meters ) zˆ  ?
20 m xˆ
2
We can also do dot products with
unit vectors. Try these:
xˆ  xˆ  ?
xˆ  yˆ  ?
1
0
yˆ  zˆ  ? 0
zˆ  zˆ  ? 1
2 xˆ  4 xˆ  ? 8
(3meters ) xˆ  (4meters ) xˆ  ?
12 m2
The calculation of work is a scalar
product or dot product:
W  F s
What is the work done by a force of 6 newtons east on an object that
is displaced 2 meters east?
12 joules
What is the work done by a force of 6 newtons east on an object that
is displaced 2 meters north?
zero
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?
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.