Exploring Vectors: Magnitude, Direction, and Operations
Vectors
(Knight: 3.1 to 3.4)
Scalars and Vectors
Temperature
= Scalar
Quantity is specified by a single
number giving its
magnitude
.
Velocity
= Vector
Quantity is specified by
three numbers that give
its
magnitude
and
direction
(or its components in three
perpendicular directions).
Properties of Vectors
Two vectors are equal if they have the same magnitude and direction.
Adding Vectors
Subtracting Vectors
Combining Vectors
Using the Tip-to-Tail Rule
Clicker Question 1
Question:
Which vector shows the sum of
A
1
+ A
2
+ A
3
?
Multiplication by a Scalar
Determining the Components of a Vector
1.
The absolute value
|
A
x
|
of the x-component
A
x
is the
magnitude of the component vector .
2.
The
sign
of
A
x
is positive if points in the positive x-direction,
negative if points in the negative x-direction.
3.
The y- and z-components,
A
y
and
A
z
, are determined similarly.
Coordinate Systems
and Vector Components
Knight’s Terminology:
•
The “x-component”
A
x
is a
scalar
.
•
The “component vector” is a
vector
that always points along the
x axis.
•
The “vector” is , and it can
point in
any
direction.
Determining Components
Cartesian and Polar
Coordinate Representations
Unit Vectors
Example:
A
= 100 i m
= 100 i m
B
= (
= (
200 Cos 45
200 Cos 45
0
0
i + 200 Cos 45
i + 200 Cos 45
0
0
j
) m
) m
= (
= (
141 i + 141
141 i + 141
j
) m
) m
Working with Vectors
C
=
A
+
B
= (
100 i m) + (-141 i + 141
100 i m) + (-141 i + 141
j
) m
) m
= (-41 i + 141
= (-41 i + 141
j
) m
) m
^
^
^
^
^
^
^
^
^
C = [C
x
2
+ C
y
2
]
½
= [
(-41 m)
(-41 m)
2
2
+ (141
+ (141
m
)
)
2
2
]
½
= 147 m
= Tan
-1
[C
y
/|C
x
|] = Tan
-1
[141/41] = 74
0
Note:
Tan
-1
ATan = arc-tangent = the angle whose tangent is …
Tilted Axes
C
x
= C Cos
C
y
= C Sin
Arbitrary Directions
Perpendicular to a Surface
Chapter 3 Summary (1)
Chapter 3 Summary (2)
Delve into the world of vectors, understanding their properties, how to add and subtract them, and determining their components in different coordinate systems. Learn about scalar quantities, vector quantities, the tip-to-tail rule, and the role of unit vectors in calculations.
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Presentation Transcript
Vectors (Knight: 3.1 to 3.4)
Scalars and Vectors Temperature = Scalar Quantity is specified by a single number giving its magnitude. Velocity = Vector Quantity is specified by three numbers that give its magnitude and direction (or its components in three perpendicular directions).
Properties of Vectors Two vectors are equal if they have the same magnitude and direction.
Clicker Question 1 Question: Which vector shows the sum of A1 + A2 + A3 ?
Coordinate Systems and Vector Components Determining the Components of a Vector 1. The absolute value |Ax| of the x-component Ax is the magnitude of the component vector . 2. The sign of Axis positive if points in the positive x-direction, negative if points in the negative x-direction. 3. The y- and z-components, Ay and Az, are determined similarly. xA x A x A Knight s Terminology: The x-component Ax is a scalar. The component vector is a vector that always points along the x axis. x A The vector is , and it can point in any direction. A
Cartesian and Polar Coordinate Representations
Unit Vectors i = = = = (1,0,0) unit vector in +x-direction = "i-hat" j (0,1,0) unit vector in +x-direction = "j-hat" = = (0,0,1) unit vector in +z-direction = "k-hat" k = + + = + + = ( , , ) A A A A A i A j A k A A A x y k x y z x y z Example: i j = + = (4, 2,5) 4 2 5 B k
Working with Vectors ^ A = 100 i m B = ( 200 Cos 450 i + 200 Cos 450 j ) m = ( 141 i + 141 j ) m ^ ^ ^ ^ C = A + B = (100 i m) + (-141 i + 141 j ) m = (-41 i + 141 j ) m ^ ^ ^ ^ C = [Cx2 + Cy2] = [(-41 m)2 + (141 m)2] = 147 m = Tan-1[Cy/|Cx|] = Tan-1[141/41] = 740 Note: Tan-1 ATan = arc-tangent = the angle whose tangent is
Tilted Axes Cx = C Cos Cy = C Sin
