Chapter 3
Scalar Quantities
Fully defined with a magnitude.
Examples:
• Time
• Temperature
• Volume
Vector Quantities
Requires both a magnitude and
direction.
Examples:
• Displacement
• Velocity
• Forces
Vector Representation
• Magnitude
• Direction
• Sense
Sense
Magnitude
Direction
F1 = 20 lb
600
F2 = 10 lb
200
Vector Addition (+>)
The net effect (resultant) of
several vectors.
Place the vectors tip-to-tail,
maintaining directions. The
resultant is the total distance
traveled.
B
A
B
R = A +> B
R
A
Graphical Method:
ü Draw vectors, to scale, using a
CAD system.
R = A +> B
A = 20 lb
B = 10 lb
200
600
R = A +> B +> C
200
A = 15 in
40
0
300
B = 12 in
C = 7 in
Analytical Method:
ü Use triangles & trigonometry
when working with two
vectors.
c = a 2 + b 2 − 2ab cos C
c
A
b
B
C
a
sin A sin B sin C
=
=
a
b
c
R = A +> B
A = 20 lb
600
B = 10 lb
200
ü Use vector components when
working with three or more
vectors.
F=30
FY
350
FX
0
S=30 20
SY
FY
sin 35 =
F
FX
cos 35 =
F
F Y = F sin 35
F X = F cos 35
SX
sin 20 =
S
SY
cos 20 =
S
S X = S sin 20
S Y = S cos 20
SX
R = A +> B +> C
200
A = 15 in
40
0
300
B = 12 in
C = 7 in
Vector Subtraction (→)
The difference between vector
quantities.
J=A→ B
Same effect as adding a negative
vector.
J = A +> (→B)
A negative vector has the same
magnitude, but opposite sense.
B
→B
A
B
→B
C
→B
J = A +> (→B)
A
J
J=A→B
B
A
subtracted vector
is placed tip-to-tip
J
K = A → B +> C
B
A
C
K
Graphical Method:
ü Place the vector being
subtracted tip-to-tip,
maintaining direction.
ü The next vector will be placed
on the tail of vector being
subtracted.
J=A→B
A = 20 lb
B = 10 lb
200
0
60
K = A → B +> C
200
A = 15 in
40
0
300
B = 12 in
C = 7 in
• Analytical Method:
Triangles.
J=A→B
A = 20 lb
B = 10 lb
200
600
Component Method.
K = A → B +> C
200
A = 15 in
40
0
300
B = 12 in
C = 7 in
Vector Equations
Equations can be written to
describe vector polygons.
B
A
D
C
A = C +> D +> B
A → B = C +> D
Write the vector equation for the
following polygon.
B
C
D
A
E
F
Vector Equations
Vectors represent magnitude and
direction.
We can solve for either:
• The magnitude & direction of
one vector
• The magnitude of two vectors.
B = 10 lb
A = 20 lb
200
0
20
C = 7 lb
600
E=?
D = 5 lb
30
0
40
0
150
F=?
A+> B +> C = D → E +> F