Chapter 3
Vectors
Prof. Raymond Lee,
revised 9-2-2010
1
• Coordinate systems
•
•
Used to describe a point’s position in space
Coordinate system consists of
•
•
•
fixed reference point called origin
specific axes with scales & labels
instructions on how to label a point relative to
origin & axes
2
• Cartesian coordinate system
•
•
•
Also called rectangular
coordinate system
x- & y- axes intersect at
origin
Points are labeled (x,y)
(compare Fig. 3-9, p. 41)
3
• Polar coordinate system
•
•
•
Origin & reference line are
noted
Point is distance r from origin
in direction of angle ! that’s
CCW from reference line
Points are labeled (r,!)
(compare Fig. 3-8, p. 41)
4
• Polar to Cartesian coordinates
•
•
•
Based on forming right
triangle from r & !
x = r cos !
y = r sin !
(compare figure on p. 43)
5
• Cartesian to polar coordinates
•
•
r is hypotenuse & ! an angle
! must be CCW from +x axis
for these equations to be
valid
6
• Cartesian example
•
•
Cartesian coordinates of a
point in xy plane are (x,y) =
(-3.50, -2.50) m, as shown in
figure. Find this point’s polar
coordinates.
Solution: From Eq. 3-6,
•
•
(SJ 2008,
p. 54)
7
• Vectors & scalars
•
•
A scalar quantity is completely specified
by a single value with an appropriate
unit & has no direction.
A vector quantity is completely
described by a number & appropriate
units + a direction.
8
• Vector notation
•
•
•
•
When handwritten, use a tilde underscore: ~
A
When printed, vector will be in boldface: A
When dealing with just the vector magnitude in
print, use italics: A or |A|
Vector magnitude has physical units & is always a
+ number
9
• Vector example
•
•
Particle travels from A to B
along path shown by dotted red
line, the scalar distance traveled
Displacement is solid line from
A to B, which is independent of
path taken between the 2
points & is a vector
(compare Fig. 3-2, p. 39)
10
• Equality of 2 vectors
•
•
•
2 vectors are equal if
they have same
magnitude & direction
A = B if A = B & they
point along parallel lines
All the vectors shown
here are equal
(compare Fig. 3-1, p. 38)
11
• Adding vectors
•
•
When adding vectors, must take into account
their directions; units must be the same
Graphical Methods
•
•
Use scale drawings
Algebraic Methods
•
More convenient
12
• Adding vectors graphically
•
•
•
Choose a scale
Draw 1st vector A with appropriate length &
in direction specified w.r.t. a coordinate
system
Draw 2nd vector with appropriate length &
in direction specified w.r.t. a coordinate
system whose origin is end of vector A & is
|| to coordinate system used for A
13
• Adding vectors graphically, 2
•
•
•
Continue drawing the
vectors “tip-to-tail”
Resultant is drawn from
tail of A to tip of last
vector
Measure length of R &
its angle
•
Use scale factor to
convert R’s length to
actual magnitude
(compare Fig. 3-3, p. 39)
14
• Adding vectors graphically, 3
•
•
If have many vectors,
repeat process until all
are included
Resultant is still drawn
from origin of 1st vector
to end of last vector
(compare Fig. 3-4, p. 39)
15
• Adding vectors, rules
•
When we add 2 vectors,
sum is independent of
order of addition.
•
•
This is commutative law of
addition
A+B=B+A
(compare Fig. 3-8, p. 39)
16
• Adding vectors, rules 2
•
When adding ! 3 vectors, sum is independent of the
way in which individual vectors are grouped
•
•
This is called associative property of addition
(A + B) + C = A + (B + C)
(Fig. 3-3, p. 39)
17
• Adding vectors, rules 3
•
•
When adding vectors, all vectors must
have same units
All vectors must measure same physical
quantity (e.g., can’t add a displacement
to a velocity)
18
• Negative of a vector
•
Negative of a vector is defined as vector
that, when added to original vector, !
resultant of zero
•
•
•
Represented as –A
A + (-A) = 0
– vector has same magnitude, but
points in opposite direction
19
• Subtracting vectors
•
•
•
Special case of vector
addition
If A – B, then use A+(-B)
Continue with standard
vector addition procedure
(compare Fig. 3-6, p. 40)
20
• Multiplying or dividing vector by a scalar
•
•
•
•
Result of multiplication or division is a vector
Vector’s magnitude is multiplied or divided by
scalar
If scalar > 0, direction of result is same as of
original vector
If scalar < 0, direction of result is opposite
that of the original vector
21
• Vector components
•
A component is a part whole
vector & is most useful with
rectangular components
•
Shown are projections of
vector along the x- & y-axes
(compare Fig. 3-8, p. 41)
22
• Vector components
•
Ax & Ay are component vectors of A
•
•
They are vectors & follow all rules for vectors
Ax & Ay are scalars & are called the
components of A
23
• Vector components, 2
•
Vector’s x-component is its projection
along x-axis
•
y-component is vector’s projection
along y-axis
•
Then
which gives
24
• Vector components, 3
•
•
y-component is moved to end
of x-component
Valid since any vector can be
moved || to itself without
being affected
•
This movement completes
triangle
(compare Fig. 3-8, p. 41)
25
• Vector components, 4
•
•
Previous equations are valid only if ! is
measured with respect to the x-axis
Components are legs of right triangle whose
hypotenuse is A
•
May still have to find " w.r.t. + x-axis
26
• Vector components, 5
•
•
Components can be +
or – & will have same
units as original vector
Components’ signs
depend on angle !
(Fig. 3.13, p. 60)
27
• Unit vectors
•
•
Unit vector is dimensionless vector with
magnitude = 1.
Use unit vectors to specify a direction &
have no other physical significance
28
• Unit vectors, 2
•
Symbols
î , ĵ, and k̂
•
represent unit vectors
They form a set of
mutually perpendicular
(#) vectors
(compare Fig. 3-13, p. 44)
29
• Unit vector notation
•
•
Ax is same as Ax , &
Ay is the same as Ay
, etc.
Write complete
vector as:
(compare Fig. 3-14,
p. 44)
30
• Adding vectors using unit vectors
•
Using R = A + B
Then
•
& so Rx = Ax + Bx & Ry = Ay + By
•
31
• Trig function warning
•
•
Component equations {Ax = A cos(!) & Ay =
A sin(!)} apply only when angle is measured
w.r.t. x-axis (preferably CCW from + x-axis).
Resultant angle {tan(") = Ay/Ax} gives angle
w.r.t. x-axis.
•
Think of triangle being formed & corresponding !;
then use appropriate trig functions
32
Adding vectors with unit vectors
(SJ 2008 Fig. 3.16, p. 61)
33
• Adding vectors using 3D unit vectors
•
Using R = A + B
•
Rx = Ax + Bx , Ry = Ay + By & Rz = Az + Bz
•
etc.
34
• Example: Taking a hike
{SJ 2008, p. 63}
•
A hiker starts a trip by first walking 25.0 km
SE from her car. She stops & sets up her tent
for the night. On 2nd day, she walks 40.0 km
in a direction 60.0° N of E, at which point she
discovers a forest ranger’s tower.
35
• hiking example, p. 2
•
(A) Determine components of
hiker’s displacement for each day.
Solution: We conceptualize problem by drawing a
sketch as in figure above. If we denote displacement
vectors on 1st & 2nd days by A & B respectively, & use
car as coordinate origin, we get vectors shown in figure.
Drawing resultant R, we can now categorize this problem
as an addition of 2 vectors.
36
• hiking example, p. 3
•
Analyze this problem using vector
components. Displacement A has
magnitude = 25.0 km & is directed
45.0° below +x axis.
From Eq. 3-5 (p. 41), its components are:
– value of Ay indicates that hiker walks in –y direction on 1st
day. Signs of Ax & Ay also are evident from figure above.
37
• hiking example, p. 4
•
2nd displacement B has a
magnitude = 40.0 km & is
60.0° N of E.
Its components are:
38
• hiking example, p. 5
•
(B) Determine the components of
the hiker’s resultant displacement
R for the trip. Find an expression
for R in terms of unit vectors.
Solution: The resultant displacement for the trip R = A + B
has components given by Eqs. 3-10 & 3-11 (p. 44):
Rx = Ax + Bx = 17.7 km + 20.0 km = 37.7 km
Ry = Ay + By = -17.7 km + 34.6 km = 16.9 km
In unit-vector form, we can write the total displacement as
R = (37.7 î + 16.9 ĵ) km
39
• hiking example, p. 6
•
Using Eq. 3-6 (p. 42), we find that
vector R has a magnitude = 41.3
km & points 24.1° N of E.
Now finalize. Units of R are km, which is reasonable for a
displacement. From graphical representation in figure, estimate
that hiker’s final position ~ (38 km, 17 km), consistent with
components of R in final result. Also, both components of R > 0,
putting final position in 1st quadrant of coordinate system, also
consistent with figure.
40
• Problem-solving strategy: Adding vectors
•
Select a coordinate system
•
•
Try to select a system that minimizes # of
components you must deal with
Draw a sketch of vectors & label each
41
• Problem-solving strategy: Adding vectors, 2
•
Find x & y components of each vector & x & y
components of resultant vector
•
•
Find z components if necessary
Use Pythagorean theorem to get resultant’s
magnitude & tangent function to get its
direction
•
Other appropriate trig functions may be needed
42