Transcript Ch3 - Vectors - Chabot College

Vectors and scalars
• A scalar quantity can be described by a single number, with
some meaningful unit
4 oranges
20 miles
5 miles/hour
10 Joules of energy
9 Volts
Vectors and scalars
• A scalar quantity can be described by a single number with
some meaningful unit
• A vector quantity has a magnitude and a direction in space, as
well as some meaningful unit.
5 miles/hour North
18 Newtons in the “x direction”
50 Volts/meter down
3-1 Vectors and Their Components


The simplest example is a displacement vector
If a particle changes position from A to B, we represent
this by a vector arrow pointing from A to B


In (a) we see that all three arrows have
the same magnitude and direction: they
are identical displacement vectors.
In (b) we see that all three paths
correspond to the same displacement
vector. The vector tells us nothing about
the actual path that was taken between A
and B.
Figure 3-1
© 2014 John Wiley & Sons, Inc. All rights reserved.
Vectors and scalars
• A scalar quantity can be described by a single number with
some meaningful unit
• A vector quantity has a magnitude and a direction in space, as
well as some meaningful unit.
• To establish the direction, you MUST first have a coordinate
system!
Standard x-y Cartesian coordinates common
Compass directions (N-E-S-W)
Drawing vectors
• Draw a vector as a line with an arrowhead at its tip.
• The length of the line shows the vector’s magnitude.
• The direction of the line shows the vector’s direction
relative to a coordinate system (that should be indicated!)
y
x
z
5 m/sec at
30 degrees from
the x axis towards y
in the xy plane
Drawing vectors
• Vectors can be identical in magnitude, direction, and units,
but start from different places…
Drawing vectors
• Negative vectors refer to direction relative to some standard
coordinate already established – not to magnitude.
Adding two vectors graphically
• Two vectors may be added graphically using either the head-to-tail
method or the parallelogram method.
Adding two vectors graphically
• Two vectors may be added graphically using either the head-to-tail
method or the parallelogram method.
Adding two vectors graphically
3-1 Vectors and Their Components

The vector sum, or resultant
o
o
Is the result of performing vector addition
Represents the net displacement of two or more
displacement vectors
Eq. (3-1)
o
Can be added graphically as shown:
Figure 3-2
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3-1 Vectors and Their Components

Vector addition is commutative
o
We can add vectors in any order
Eq. (3-2)
Figure (3-3)
© 2014 John Wiley & Sons, Inc. All rights reserved.
3-1 Vectors and Their Components

Vector addition is associative
o
We can group vector addition however we like
Eq. (3-3)
Figure (3-4)
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3-1 Vectors and Their Components

A negative sign reverses vector
direction
Figure (3-5)

We use this to define vector
subtraction
Eq. (3-4)
Figure (3-6)
© 2014 John Wiley & Sons, Inc. All rights reserved.
3-1 Vectors and Their Components


These rules hold for all vectors, whether they
represent displacement, velocity, etc.
Only vectors of the same kind can be added
o
(distance) + (distance) makes sense
o
(distance) + (velocity) does not
© 2014 John Wiley & Sons, Inc. All rights reserved.
3-1 Vectors and Their Components


These rules hold for all vectors, whether they
represent displacement, velocity, etc.
Only vectors of the same kind can be added
o
(distance) + (distance) makes sense
o
(distance) + (velocity) does not
Answer:
(a) 3 m + 4 m = 7 m
(b) 4 m - 3 m = 1 m
© 2014 John Wiley & Sons, Inc. All rights reserved.
Adding more than two vectors graphically
• To add several vectors, use the head-to-tail method.
• The vectors can be added in any order.
Adding more than two vectors graphically—Figure 1.13
• To add several vectors, use the head-to-tail method.
• The vectors can be added in any order.
Subtracting vectors
• Reverse direction, and add normally head-to-tail…
Subtracting vectors
Multiplying a vector by a scalar
• If c is a scalar,
the

product cA has
magnitude |c|A.
Addition of two vectors at right angles
• First add vectors graphically.
• Use trigonometry to find magnitude & direction of sum.
Addition of two vectors at right angles
• Displacement (D) = √(1.002 + 2.002) = 2.24 km
• Direction f = tan-1(2.00/1.00) = 63.4º East of North
Note how the final answer has THREE things!
• Answer: 2.24 km at 63.4 degrees East of North
• Magnitude (with correct sig. figs!)
Note how the final answer has THREE things!
• Answer: 2.24 km at 63.4 degrees East of North
• Magnitude (with correct sig. figs!)
• Units
Note how the final answer has THREE things!
• Answer: 2.24 km at 63.4
degrees East of North
• Magnitude (with correct sig. figs!)
• Units
• Direction
Components of a vector
• Represent any vector by an x-component Ax and a y-component Ay.
• Use trigonometry to find the components of a vector: Ax = Acos θ and
Ay = Asin θ, where θ is measured from the +x-axis toward the +y-axis.
Positive and negative components
• The components of a vector can be
positive or negative numbers.
Finding components
• We can calculate the components of a vector from its magnitude and
direction.
3-1 Vectors and Their Components
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

Components in two dimensions can be found by:
Where θ is the angle the vector makes with the
positive x axis, and a is the vector length
The length and angle can also be found if the
components are known
© 2014 John Wiley & Sons, Inc. All rights reserved.
3-1 Vectors and Their Components

In the three dimensional case we need more
components to specify a vector
o
(a,θ,φ) or (ax,ay,az)
© 2014 John Wiley & Sons, Inc. All rights reserved.
3-1 Vectors and Their Components

In the three dimensional case we need more
components to specify a vector
o
(a,θ,φ) or (ax,ay,az)
Answer: choices (c), (d), and (f) show the components properly arranged to
form the vector
© 2014 John Wiley & Sons, Inc. All rights reserved.
Calculations using components
• We can use the components of a vector to find its magnitude
Ay
and direction:
2
2
A A  A
and tan 
x
y
• We can use the components of a
set of vectors to find the components
of their sum:
Rx  Ax  Bx  Cx  , Ry  Ay  By  Cy 
Ax
Adding vectors using their components
Unit vectors
• A unit vector has a magnitude
of 1 with no units.
• The unit vector î points in the
+x-direction, jj points in the +ydirection, and kk points in the
+z-direction.
• Any vector can be expressed
in terms of its components as

A =Axî+ Ay jj + Az kk.
3-2 Unit Vectors, Adding Vectors by Components
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
A unit vector
o
Has magnitude 1
o
Has a particular direction
o
Lacks both dimension and unit
o
Is labeled with a hat: ^
We use a right-handed coordinate system
o
Remains right-handed when rotated
© 2014 John Wiley & Sons, Inc. All rights reserved.
3-2 Unit Vectors, Adding Vectors by Components



The quantities axi and ayj are vector components
The quantities ax and ay alone are scalar
components
Vectors can be added using components
Eq. (3-9)
→
© 2014 John Wiley & Sons, Inc. All rights reserved.
3-2 Unit Vectors, Adding Vectors by Components

To subtract two vectors, we subtract components
Eq. (3-13)
Unit Vectors, Adding Vectors by Components
© 2014 John Wiley & Sons, Inc. All rights reserved.
3-2 Unit Vectors, Adding Vectors by Components

To subtract two vectors, we subtract components
Eq. (3-13)
Unit Vectors, Adding Vectors by Components
Answer: (a) positive, positive
(b) positive, negative
(c) positive, positive
© 2014 John Wiley & Sons, Inc. All rights reserved.
3-3 Multiplying Vectors

Multiplying two vectors: the scalar product
o
o
Also called the dot product
Results in a scalar, where a and b are magnitudes and φ is
the angle between the directions of the two vectors:
Eq. (3-20)

The commutative law applies, and we can do the dot
product in component form
Eq. (3-22)
Eq. (3-23)
© 2014 John Wiley & Sons, Inc. All rights reserved.
3-3 Multiplying Vectors

A dot product is: the product of the magnitude of one
vector times the scalar component of the other vector
in the direction of the first vector
Eq. (3-21)
Figure (3-18)


Either projection of one
vector onto the other can
be used
To multiply a vector by the
projection, multiply the
magnitudes
© 2014 John Wiley & Sons, Inc. All rights reserved.
3-3 Multiplying Vectors
Answer: (a) 90 degrees (b) 0 degrees (c) 180 degrees
© 2014 John Wiley & Sons, Inc. All rights reserved.
The scalar product
The scalar
product of two
vectors (the “dot
product”) is
A · B = ABcosf
The scalar product
The scalar
product of two
vectors (the “dot
product”) is
A · B = ABcosf
The scalar product
The scalar product of two vectors (the “dot product”) is
A · B = ABcosf
Useful for
•Work (energy) required or released as force is applied over
a distance (4A)
•Flux of Electric and Magnetic fields moving through
surfaces and volumes in space (4B)
Calculating a scalar product
By components, A · B = AxBx + AyBy + AzBz
Example: A = 4.00 m @ 53.0°, B = 5.00 m @ 130°
Calculating a scalar product
By components, A · B = AxBx + AyBy + AzBz
Example: A = 4.00 m @ 53.0°, B = 5.00 m @ 130°
Ax = 4.00 cos 53 = 2.407
Ay = 4.00 sin 53 = 3.195
Bx = 5.00 cos 130 = -3.214
By = 5.00 sin 130 = 3.830
AxBx + AyBy = 4.50 meters
A · B = ABcosf  (4.00)(5.00) cos(130-53) = 4.50 meters2
The vector product
• The vector
product (“cross
product”) of two
vectors has
magnitude
| A B |  ABsinf
and the righthand rule gives
its direction.
3-3 Cross Products
The cross product of two vectors with magnitudes a & b,
separated by angle φ, produces a vector with magnitude:
o
Eq. (3-24)
And a direction perpendicular to both original vectors


Direction is determined by the right-hand rule
Place vectors tail-to-tail, sweep fingers from the first to
the second, and thumb points in the direction of the
resultant vector
© 2014 John Wiley & Sons, Inc. All rights reserved.
3-3 Multiplying Vectors
Figure (3-19)
The upper shows vector a cross vector b, the lower shows vector b cross vector a
© 2014 John Wiley & Sons, Inc. All rights reserved.
The vector product
•The vector product (“cross product”) A x B of two vectors is a vector
•Magnitude = AB sin f
•Direction = orthogonal (perpendicular) to A and B,
using the “Right Hand Rule”
y
x
B
z
A
AxB
3-3 Multiplying Vectors

The cross product is not commutative
Eq. (3-25)

To evaluate, we distribute over components:
Eq. (3-26)

Therefore, by expanding (3-26):
Eq. (3-27)
© 2014 John Wiley & Sons, Inc. All rights reserved.
3-3 Multiplying Vectors
© 2014 John Wiley & Sons, Inc. All rights reserved.
3-3 Multiplying Vectors
Answer: (a) 0 degrees (b) 90 degrees
© 2014 John Wiley & Sons, Inc. All rights reserved.
The vector cross product
The cross product of two vectors is
A x B (with magnitude ABsinf)
Useful for
•Torque from a force applied at a distance away from an
axle or axis of rotation (4A)
•Calculating dipole moments and forces from Magnetic
Fields on moving charges (4B)
3-1 Vectors and Their Components

Angles may be measured in degrees or radians

Recall that a full circle is 360˚, or 2π rad

Know the three basic trigonometric functions
Figure (3-11)
© 2014 John Wiley & Sons, Inc. All rights reserved.
3-2 Unit Vectors, Adding Vectors by Components


Vectors are independent of the
coordinate system used to measure
them
We can rotate the coordinate
system, without rotating the vector,
and the vector remains the same
Eq. (3-14)
Eq. (3-15)

All such coordinate systems are
equally valid
Figure (3-15)
© 2014 John Wiley & Sons, Inc. All rights reserved.
3
Summary
Scalars and Vectors
Adding Geometrically

Scalars have magnitude only

Vectors have magnitude and
direction


Eq. (3-2)
Eq. (3-3)
Both have units!
Vector Components

Unit Vector Notation
Given by

Eq. (3-5)

Obeys commutative and
associative laws
We can write vectors in terms
of unit vectors
Eq. (3-7)
Related back by
Eq. (3-6)
© 2014 John Wiley & Sons, Inc. All rights reserved.
3
Summary
Adding by Components

Add component-by-component
Eqs. (3-10) - (3-12)
Scalar Product

Scalar Times a Vector

Product is a new vector

Magnitude is multiplied by
scalar

Direction is same or opposite
Cross Product
Dot product
Eq. (3-20)

Produces a new vector in
perpendicular direction

Direction determined by righthand rule
Eq. (3-22)
© 2014 John Wiley & Sons, Inc. All rights reserved.
Eq. (3-24)