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MECHANICS
Introduction to Vector Math
Scalar
A scalar is a physical quantity that is completely described by a real number. Scalars may have
units, but they have no direction. Scalars just “are.”
The following are scalars, physical quantities that are important in engineering and design:
Time. 3:00 PM, 6 minutes (m), 11 hours (h), 1 decade, etc., are examples of time.
The magnitude or size of time is a real number. There are units (minutes, hours, etc.).
However, there is no direction. Hours do not move up, minutes do not go east, 3:00 PM
does not point to the left. Time is a scalar.
Mass. 26 kilograms (kg), 19 slugs, 8 grams (g), etc., are examples of mass. Mass is a
measure of the amount of material that makes up an object. These quantities are real
numbers as mass is real; they have units, depending on whether one works in the U. S.
customary system or metric system. However, there is no direction. Grams do not point
up or down, the U. S. customary system unit for mass, slugs, does not point to the east.
Mass describes physical matter, but does not have a direction; it is a scalar.
Temperature. 22° Fahrenheit (F), -5° Celsius (C), 268° Kelvin (K), etc., are examples of
temperature. As with the other scalars, there is no direction associated with temperature,
it just describes a physical, real, measurable quantity with units.
There are many other examples of scalars as they occur often in the physical world. The key
idea to remember is that they may have units, but no direction.
Vector
A vector, on the other hand, is used to completely describe a physical quantity that
requires magnitude and direction. Vectors are used to describe forces, positions, velocities,
accelerations, moments or couples, momentum, etc.
Force. 120 pounds (lb), 534 Newtons (N), 8 tons, etc., are examples of force. There is a
magnitude of force, and units. Additionally, there is direction. An object that weighs 8 tons
is associated with a vector that points towards the core of the earth, that is the direction in
which the force lies.
Throwing a ball to the right imparts a force of, say, 10 N, on the ball directed to the right.
Force always has magnitude, direction, and units.
→
W
→
F
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MECHANICS
Introduction to Vector Math
Vector
continued
Position. 82 feet (ft), 25 meters (m), 0.013 nautical miles (nm),
etc., are examples of position. Position is always measured from
a reference point. It could be measured from an absolute point,
such as the distance from a landmark (e.g., 20 miles (mi) west of
the Steel Building), or relative point (e.g., Stan is 2 m to the left of
Dave). Since a reference point is involved, position has a direction
from that point, and is measured in real numbers with units.
Position is a vector.
Object
Reference
Point
Velocity. 25 ft/s, 7.62 m/s, 17 mi/h, etc., are examples of velocity.
When objects move, they move in a direction, and velocity is a
vector that points in the direction of movement. If a ball is thrown
upward with a speed of 20 mi/h, the velocity vector points upward.
Velocity is a vector with magnitude, direction and units.
→
v
Vectors occur very often in the physical world to describe physical quantities that have
magnitude and direction.
Vector Notation
Scalars are simply noted as letters, such as m, t, μ, d, etc.
→
→
→
→
→
Vectors are denoted as a letter with an arrow above it. F , s, v , a, and M are all vectors.
Often one may wish to use, compute, or discuss just the magnitude portion of a vector quantity.
An example may be to discuss the speed of an object; speed being another name for the
magnitude of velocity. To denote the magnitude of a vector, the options are to reuse the letter
with no arrow above it, or to use an absolute value notation around the symbol. For example,
→
to denote the magnitude of velocity (speed) use v or v .
Graphically, a vector is an arrow that points in the direction of the physical quantity it is
representing. The length of the arrow is proportional to the quantities magnitude. For example,
10 lb of force (F), and 15 lb of force (G), acting in the same direction
→
→
G
F
Or a displacement of 10 ft to the NE (s), or 10 ft due west (r)
LESSONS MECHANICS
→
→
s
r
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VEX Curriculum 1.0 © 2008
MECHANICS
Introduction to Vector Math
Rules for Manipulating Vectors
Vector Equality
Two vectors are equal if and only if their magnitudes and directions are equal.
→
→
→
H
J
R
→
→
S
→
→
→
R ≠S
H =J
→
→
K
L
→
→
K =L
Vector Addition
→
→
→
→
A
→
B
C = A+ B
→
C
The Triangle Rule for vector addition says when you place vectors head to tail, their sum
is independent of the order in which they are placed together. Each vector must maintain its
direction and magnitude.
Consider
→
→
A
→
→
B
→
Then A+ B equals C
→
→
C
→
B
LESSONS MECHANICS
→
A
→
B
→
A
C
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Introduction to Vector Math
Vector Addition continued
For mathematical operations, vectors can be moved anywhere, as long as their direction
and magnitude are preserved. The Parallelogram Rule allows us to add vectors tail to tail,
providing another graphical means of adding vectors.
Consider
→
→
→
→
A
A
A
C
→
→
→
B
B
B
The direct application of the Parallelogram Rule is when one must break a vector down
into parts, or components, for ease of analysis. Given any vector can be decomposed into
components parallel to any frame of reference one desires to use.
y
y
→
F
x
→
→
Fy
F
x
→
Fx
→
→
→
→
Vector addition is commutative A+ B = B + A
Vector addition is associative
LESSONS MECHANICS
⎛→ →⎞ → → ⎛→ →⎞
⎜ A+ B ⎟ + C = A+ ⎜ B + C ⎟
⎝
⎠
⎝
⎠
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Introduction to Vector Math
Product of a Scalar and a Vector
When one multiplies a positive scalar (a real number) times a vector (with magnitude and
direction), the result as another vector. This resultant vector retains the direction of the original
vector, but its magnitude is changed by the product of the scalar. A vector that represents
10 ft/s, when multiplied by a scalar of 3, becomes a vector that points in the original direction of
the velocity, but has a magnitude of 30 ft/s.
→
h
→
v
→
→
h = 3v
Similarly, when one multiplies a negative scalar times a vector, the result as another vector.
This resultant vector retains the same line of action, but points in the opposite direction of
the original vector. As before, the magnitude is changed by the product of the scalar. A vector
that represents 10 ft/s, when multiplied by a scalar of -3, becomes a vector that points in the
opposite direction of the original vector, and has a magnitude of 30 ft/s.
→
h
→
v
→
→
h = −3 v
Quotient of a Scalar and a Vector
When one divides a positive scalar into a vector, the result as another vector that retains the
direction of the original vector, but its magnitude is changed by the quotient of the scalar.
Similarly, when one divides a negative scalar into a vector, the resultant vector’s magnitude
changes with the quotient, and it points in the opposite direction of the original vector.
→
v
→
→
→
1
h= v
3
h
→
v
→
→
→
1
h=− v
3
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MECHANICS
Introduction to Vector Math
Quotient of a Scalar and a Vector continued
Scalar-Vector product is associative
→
⎛ →⎞
a ⎜ bU ⎟ = (ab )U
⎝
⎠
Scalar-Vector product is distributive
→
→
→
with respect to scalar addition
(a + b )U = aU + bU
with respect to vector addition
→
→
⎛→ →⎞
a ⎜ U + V ⎟ = aU + aV
⎝
⎠
Vector Subtraction
→
→
Consider the original vectors A and B
→
→
→
A
B
→
To determine A− B , multiply the vector by -1,
→
and treat the problem like vector addition.
A
→
(−1)B
→
−B
→
→
→
→
→
Then D = A− B = A+ (−1)B
→
D
→
A
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VEX Curriculum 1.0 © 2008
MECHANICS
Introduction to Vector Math
Unit Vectors
A Unit Vector is a vector with direction, but with a magnitude of 1. Meaning, regardless of the
system you are working in, whether forces, velocities, etc., a unit vector will be 1 unit long.
Unit vectors have a specific notation.
Rather than an arrow over the letter,
they have a carrot.
∧
e
Unit vectors are used to preserve a regular vector’s direction while manipulating its magnitude.
Their use will be apparent with further study in mechanics.
Unit vectors are constructed by dividing a vector by its own magnitude.
→
∧
F
e=
F
For example, one is working in a horizontal/vertical frame of reference. A position vector points
to the right, parallel to the horizontal axis, and is 10 ft long. If one wanted to construct a unit
vector that would preserve the direction of the position vector, one would, merely, divide the
position vector by its own length.
∧
10 i ft ∧
=i
r = 10 i ft. Unit vector e would be constructed as e =
10 ft
→
∧
∧
∧
∧
∧
⇒ e=i
Note, dividing the vector by its own magnitude causes one to divide out the units, so a unit
vector is 1 unit length long, and has a specific direction, but no units.
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