Transcript Part II

Analytic Method of Vector Addition
Analytic Method of Addition
• First: Discuss the resolution of
vectors into components:
YOU MUST KNOW &
UNDERSTAND
TRIGONOMETERY TO
UNDERSTAND THIS!!!!
Vector Components
• Consider a vector V in a plane (say, the xy plane)
• We can express V in terms of
Components Vx , Vy
• Finding components Vx & Vy is equivalent to
finding 2 mutually perpendicular vectors which,
when added (with vector addition) will give V.
• That is, find Vx & Vy such that
V  Vx + Vy
(Vx || x axis, Vy || y axis)
Finding components
 “Resolving into components”
• Mathematically, a component is a
projection of a vector along an axis
– Any vector can be
completely described
by its components
• It is useful to use
rectangular components
– These are the projections
of the vector along the xand y-axes
When V is resolved into components:
Vx & V y
V  Vx + Vy (Vx || x axis, Vy || y axis)
By the parallelogram method, the vector sum is:
V = Vx + Vy
In 3 dimensions, we also need a component Vz
Brief Trig Review
• Adding vectors in 2 & 3 dimensions using
components requires
TRIGONOMETRY FUNCTIONS
HOPEFULLY, A REVIEW!!
See also Appendix A!!
• Given any angle θ, we can
construct a right triangle:
Hypotenuse  h
Adjacent side  a
Opposite side  o
h
a
o
• Define the trigonometry functions in terms
of h, a, o:
= (opposite side)/(hypotenuse)
= (adjacent side)/(hypotenuse)
= (opposite side)/(adjacent side)
[Pythagorean theorem]
Signs of Sine, Cosine, Tangent
Trig identity: tan(θ) = sin(θ)/cos(θ)
Using Trig Functions to Find Vector Components
We can & we will use
all of this to
Add Vectors
Analytically!
[Pythagorean theorem]
Example
V = Displacement 500 m, 30º N of E
Unit Vectors
• Its convenient to express vector A in terms of it’s components
Ax, Ay, Az & UNIT VECTORS along x,y,z axes
UNIT VECTOR  a dimensionless vector, length = 1
• Define unit vectors along the x,y,z axes:
i along x; j along y; k along z
|i| = |j| = |k| = 1
Figure 
• Example: Vector A in the x-y plane.
Components Ax, Ay:
A  Axi + Ayj
Figure 
Simple Example
• Position vector r in the x-y plane.
Components x, y:
r  xi + y j
Figure 
Vector Addition Using Unit Vectors
• Suppose we want to add two vectors V1 & V2 in
the x-y plane:
V = V1 + V2
“Recipe”
1. Find x & y components of V1 & V2 (using trig!)
V1 = V1xi + V1yj
V2 = V2xi + V2yj
2. x component of V: Vx = V1x + V2x
y component of V: Vy = V1y + V2y
3. So
V = V1 + V2 = (V1x+ V2x)i + (V1y+ V2y)j
Using Components to Add Two Vectors
• Consider 2 vectors, V1 & V2. We want V = V1 + V2
• Note: The components of each vector are really onedimensional vectors, so they can be added arithmetically.
We want the vector sum V = V1 + V2
“Recipe” (for adding 2 vectors using trig & components)
1. Sketch a diagram to roughly add the vectors graphically.
Choose x & y axes.
2. Resolve each vector into x & y components using sines & cosines.
That is, find V1x, V1y, V2x, V2y. (V1x = V1cos θ1, etc.)
4. Add the components in each direction. (Vx = V1x + V2x, etc.)
5. Find the length & direction of V, using:
Example
A rural mail carrier leaves the post office & drives
22.0 km in a northerly direction. She then drives in
a direction 60.0° south of east for 47.0 km. What is
her displacement from the post office?
A rural mail carrier leaves the
post office & drives 22.0 km in a
northerly direction. She then
drives in a direction 60.0° south
of east for 47.0 km. What is her
displacement from the post
office?
Solution
NOTE!!!!
Example
A plane trip involves 3 legs, with 2 stopovers: 1) Due east
for 620 km, 2) Southeast (45°) for 440 km, 3) 53° south of
west, for 550 km. Calculate the plane’s total displacement.
A plane trip involves 3 legs, with 2
stopovers: 1) Due east for 620 km,
2) Southeast (45°) for 440 km,
3) 53° south of west, for 550 km.
Calculate the plane’s total
displacement.
Solution
Another Analytic Method
• Laws of Sines & Law of Cosines from trig.
• Arbitrary triangle:
• Law of Cosines: c2 = a2 + b2 - 2 a b cos(γ)
• Law of Sines: sin(α)/a = sin(β)/b = sin(γ)/c
• Add 2 vectors: C = A + B
• Law of Cosines:
C2 = A2 + B2 -2 A B cos(γ)
Gives length of resultant C.
• Law of Sines:
sin(α)/A = sin(γ)/C, or sin(α) = A sin(γ)/C
Gives angle α