Scalars and Vectors
Scalars and Vectors
A scalar is a number which expresses quantity. Scalars may or may not have units associated with them.
Examples: mass, volume, energy, money
A vector is a quantity which has both magnitude and direction. The magnitude of a vector is a scalar.
Examples: Displacement, velocity, acceleration, electric field
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Vector Notation
• Vectors are denoted as a symbol with an arrow over the top: x
• Vectors can be written as a magnitude and direction:
= 15 .
7 N C @ 30 o deg
Vector Representation
• Vectors are represented by an arrow pointing in the direction of the vector.
• The length of the vector represents the magnitude of the vector.
• WARNING!!! The length of the arrow does not necessarily represent a length.
= 2 .
3 m s
2

Vector Addition
Adding Vectors Graphically.
Arrange the vectors in a head to tail fashion.
= +
The resultant is drawn from the tail of the first to the head of the last vector.
Vector Addition
This works for any number of vectors.
= + + +
3

Vector Addition
Vector Subtraction
Subtracting Vectors Graphically.
−
Flip one vector.
Then proceed to add the vectors
= − B
B
= +
The resultant is drawn from the tail of the first to the head of the last vector.
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Vector Components
Any vector can be broken down into components along the x and y axes.
r r x
Example: = components.
5 .
0 m @ 30 o from the horizontal. Find its
= r r
= r r x
+ r r
θ r cos θ i
ˆ r r r y r y
= r sin θ ˆ j r r r r x r r y r r y x
=
=
=
=
(
4
5 .
.
3
0 m m i
ˆ
(
5 .
0 m
)
2 .
5 m
ˆ j
) cos sin
30 o i
ˆ
30 o
ˆ j
Vector Addition by Components
You can add two vectors by adding the components of the vector along each direction. Note that you can only add components which lie along the same direction.
+
+
= 3 .
2 m
=
=
1 .
5
4 .
7 m m s i ˆ + 2 .
5 m s i
ˆ + 5 .
2 m s i
ˆ + 7 .
7 m s ˆ j s
ˆ j s
ˆ j
+ B = 12 .
4 m s
Never add the x-component and the y-component
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Unit Vectors
Unit vectors have a magnitude of 1.
They only give the direction.
ˆ j y i
ˆ x
A displacement of 5 m in the x-direction is written as d r
= 5 m i ˆ
The magnitude is 5m.
The direction is the î-direction.
k
ˆ z
Finding the Magnitude and Direction
Pythagorean Theorem r = r r
θ r r x r x
2 + r y
2 r r y tan θ = r y r x
θ = tan − 1

 r y r x


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Vector Multiplication I: The Dot Product
The result of a dot product of two vectors is a scalar !
r
B
AB cos
θ i
ˆ ⋅ i
ˆ = 1 j
ˆ ⋅ j
ˆ = 1 k
ˆ ⋅ k
ˆ = 1 i
ˆ ⋅ j
ˆ = 0 j
ˆ ⋅ k
ˆ = 0 i
ˆ ⋅ k
ˆ = 0
Vector Multiplication I: The Dot Product
=
(
2 i
ˆ + 3 j
ˆ − 2 k
ˆ
)
N s r
=
(
3 i
ˆ − 4 j
ˆ − 6 k
ˆ
) m
⋅ s r
= 2 ( 3 ) N ⋅ m + 3 ( − 4 ) N ⋅ m + ( − 2 )( − 6 ) N ⋅ m
s r
6 N
m
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Vector Multiplication II: The Cross Product
The result of a cross product of two vectors is a new vector !
A
B
AB sin
θ i
ˆ × i
ˆ = 0 j
ˆ × ˆ j = 0 k
ˆ × k
ˆ = 0 i
ˆ × j
ˆ = k
ˆ × i
ˆ = k
ˆ j × k
ˆ = i
ˆ
ˆ j
ˆ
Vector Multiplication II: The Cross Product q
( ) ( q v r
×
) (
× q r )
=
(
×
) (
×
)
θ
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Vector Multiplication II: The Cross Product
=
(
2 i
ˆ + 3 j
ˆ − 2 k
ˆ
)
N
τ v
= r r
( r r
×
)
=
(
3 i
ˆ − 4 j
ˆ − 6 k
ˆ
) m
τ v
= 3 m i
( )
+
( )(
− 4 m
) i
( )
+
( )(
− 6 m
) i
ˆ k
ˆ
+
+
( )
+
( )(
− 4 m
) ( )
+
( )(
− 6 m
) ˆ j k
ˆ
(
− 2 N
)( ) k
( )
+
(
− 2 N
)(
− 4 m
)
( ) j +
(
− 2 N
)(
− 6 m
) k
( )
τ v
=
(
26 i ˆ − 6 j ˆ + 17 k
ˆ
)
N ⋅ m




Vector Multiplication II: The Cross Product
−
3
= j ˆ k
ˆ
4
− k
ˆ
6
− 2
(
2 i
ˆ
−
−
3
2 i
ˆ
6
2


+


3 j
ˆ − 2 k
ˆ
τ v
)
N
= i ˆ j ˆ


3
2
−
3
4 

τ v
τ v r r
( r r
×
)
=
=
(
=
+
(
(
−
+
3 i
ˆ
(
−
26 i
ˆ
−
−
4
6 j
ˆ j
ˆ
−
+
6 k
ˆ
17
)
4
( ) ( )( ) ) i
ˆ
6
( ) ( )( ) ) j
ˆ
(
3
( ) ( )( ) ) k
ˆ k
ˆ
) m
N ⋅ m
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Vector Multiplication II: Right Hand Rule
Index finger in the direction of the first vector.
Middle finger in the direction of the second vector
Thumb points in the direction of the cross product.
WARNING: Make sure you are using your right hand!!!
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