Vectors in Physics Walker Ch 3

1
Vectors in Physics(Walker Ch 3)
North
2 km away
?
?
v1
East
v2
?
2
Vectors and Scalars Walker Ch. 3 (3-1)

• A scalar is a quantity that is completely
  specified by a positive or negative number with
  an appropriate unit and has no direction.
• A vector is a physical quantity that must be
  described by a magnitude (number) and appropriate
  units plus a direction.

3
Some Notes About Scalars

• Some examples
• Temperature (of 20o C)
• Volume (of 45 cm3)
• Mass (of 5.7 kg)
• Time intervals (of 24 h)
• Rules of ordinary arithmetic are used to
  manipulate scalar quantities

4
Vector Example

• A particle travels from A to B along the path
  shown by the dotted red line
• This is the distance traveled and is a scalar
• The displacement is the solid line from A to B
• The displacement is independent of the path taken
  between the two points
• Displacement is a vector
• Notice the arrow indicating direction

5
Other Examples of Vectors

• Displacement (of 3.5 km at 20o North of East)
• Velocity (of 50 km/h due North)
• Acceleration (of 9.81 m/s2 downward)
• Force (of 10 Newtons in the x direction)

6
Vector Notation

• When handwritten, use an arrow
• When printed, will be in bold print with an
  arrow
• When dealing with just the magnitude of a vector
  in print, an italic letter will be used A or
  
• The magnitude of the vector has physical units
• The magnitude of a vector is always a positive
  number

7
Some Properties of Vectors
Equality of Two Vectors

• Two vectors are equal if they have the same
  magnitude and the same direction
• if and they are
  parallel and point in the same direction
• All of the vectors shown are equal

8
Adding Vectors (Sect 3-3)

• When adding vectors, their directions must be
  taken into account
• Units must be the same
• Graphical Methods
• Use scale drawings
• Accuracy difficult to control
• Algebraic Methods
• Accuracy well defined

9
Rules for Adding Vectors Graphically

• Draw the first vector with the appropriate
  length and direction
• Draw the next vector with the appropriate
  length and direction specified, whose origin is
  located at the end of vector
• Continue drawing the vectors tip-to-tail
• The resultant is drawn from
  the origin of to the end of
  the last vector
• Measure the length
  and angle of

10
Adding Vectors Graphically, cont.

• When you have many vectors, just keep repeating
  the process until all are included
• The resultant is still drawn from the origin of
  the first vector to the end of the last vector

11
Adding Vectors Graphically, final

• Example A car travels 3 km North, then 2 km
  Northeast, then 4 km West, and finally 3 km
  Southeast. What is the resultant displacement?

R
R is 2.4 km, 13.5 W of N or 103.5º from ve
x-axis.
12
Components of a Vector (Walker Sect. 3-2)
Components of a Vector

• A component is a part
• It is useful to use rectangular components
• These are the projections of the vector along the
  x- and y-axes

13
Vector Component Terminology

• are the component vectors
  of
• They are vectors and follow all the rules for
  vectors
• Ax and Ay are scalars, and will be referred to as
  the components of
• The combination of the component vectors is a
  valid substitution for the actual vector

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Components of a Vector, 2

• The x-component of a vector is the projection
  along the x-axis
• The y-component of a vector
  is the projection along the y-axis
• When using this form of the equations, q must be
  measured ccw from the positive x-axis
  (mathematical standard definition)

15
Components of a Vector, 3

• The y-component is moved to the end of the
  x-component
• This is due to the fact that any vector can be
  moved parallel to itself without being affected
• This completes the triangle

16
Components of a Vector, 4.

• The components are the legs of the right triangle
  whose hypotenuse is
• Must find ? with respect to the positive x-axis
• Use the signs of Ax and Ay and a sketch to track
  the correct value of ?

The Pythagorean theorem
17
Components of a Vector, 5

• The components can be positive or negative and
  will have the same units as the original vector
• The signs of the components will depend on the
  angle

18
Components of a Vector

• Example 1. Find Ax and Ay for the vector A with
  magnitude and direction given by A 3.5 m and
  ?66o respectively. (Ex. 3-1 p. 57)
• Example 2. Ax -0.50m and Ay 1.0 m. Find the
  direction of the vector A, and its magnitude.

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Components of a Vector, final

• Cx Ax Bx
• Cy Ay By

20
Components of a Vector, Example

• Example A car travels 3 km North, then 2 km
  Northeast, then 4 km West, and finally 3 km
  Southeast. What is the resultant displacement?
  Use the component method of vector addition.

N
y
B
A
By
C
x
Bx
W
E
Dx
Dy
D
S
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Components of a Vector, Example
Rx Ax Bx Cx Dx 0 km 1.4 km - 4.0 km
2.1 km -0.5 km Ry Ay By Cy Dy 3.0 km
1.4 km 0 km - 2.1 km 2.3 km
Magnitude
N
y
R
Ry
Direction
x
E
W
Rx
Stop. Think. Is this reasonable? NO! Off by
180º. Answer -78º 180 102 from ve x-axis.
S
22
Adding Vectors, Rules

• When two vectors are added, the sum is
  independent of the order of the addition.
• This is the commutative law of addition

23
Adding Vectors, Rules final

• When adding vectors, all of the vectors must have
  the same units
• All of the vectors must be of the same type of
  quantity
• For example, you cannot add a displacement to a
  velocity

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Negative of a Vector

• The negative of a vector is defined as the vector
  that, when added to the original vector, gives a
  resultant of zero
• Represented as
• The negative of the vector will have the same
  magnitude, but point in the opposite direction

25
Subtracting Vectors

• Special case of vector addition
• Continue with standard vector addition procedure

26
Multiplying or Dividing a Vector by a Scalar

• The result of the multiplication or division is a
  vector
• The magnitude of the vector is multiplied or
  divided by the scalar
• If the scalar is positive, the direction of the
  result is the same as of the original vector
• If the scalar is negative, the direction of the
  result is opposite that of the original vector

27
Multiplying or Dividing a Vector by a Scalar

• Example Given

Multiply by 2
Divide by -2
2A
-(1/2)A
28
Multiplying Vectors

• Two vectors can be multiplied in two different
  ways
• One is the scalar product (result is a scalar)
• Also called the dot product
• The other is the vector product (result is a
  vector)
• Also called the cross product
• These products will be discussed as they arise in
  applications