trigonometric method of adding vectors. analytic method of addition resolution of vectors into...
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Trigonometric Method of Adding Vectors
Analytic Method of Addition Resolution of vectors into components:
YOU MUST KNOW & UNDERSTAND
TRIGONOMETERY TO
UNDERSTAND THIS!!!!
Vector Components • Any vector can be expressed as the sum of
two other vectors, called its components. Usually, the other vectors are chosen so that they are perpendicular to each other.
• Consider the vector V in a plane (say, the xy plane)
• We can express V in terms ofCOMPONENTS Vx , Vy
• Finding THE COMPONENTS Vx & Vy is EQUIVALENT to finding 2 mutually perpendicular vectors which, when added (with vector addition) will give V.
• We can express any vector V in terms of
COMPONENTS Vx , Vy • Finding Vx & Vy is EQUIVALENT to
finding 2 mutually perpendicular vectors which, when added (with vector addition) will give V.
• That is, we want to find Vx & Vy such thatV 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 useRectangular Components
–These are the projections of the vector along the x- and y-axes
V is Resolved Into Components: Vx & Vy
V Vx + Vy (Vx || x axis, Vy || y axis)
By the parallelogram method, clearlyTHE VECTOR SUM IS: V = Vx + Vy
In 3 dimensions, we also need a Vz.
Brief Trig Review
Hypotenuse h, Adjacent side aOpposite side o
ho
a
• Adding vectors in 2 & 3 dimensions using components requires TRIG FUNCTIONS
• HOPEFULLY, A REVIEW!! – See also Appendix A!!
• Given any angle θ, we can construct a right triangle:
• Define the trig functions in terms of h, a, o:
= (opposite side)/(hypotenuse)
= (adjacent side)/(hypotenuse)
= (opposite side)/(adjacent side)
[Pythagorean theorem]
Trig Summary
• Pythagorean Theorem: r2 = x2 + y2
• Trig Functions: sin θ = (y/r), cos θ = (x/r) tan θ = (y/x)• Trig Identities: sin² θ + cos² θ = 1• Other identities are in Appendix B & the back cover.
Signs of the Sine, Cosine & Tangent Trig Identity: tan(θ) = sin(θ)/cos(θ)
Inverse Functions and Angles• To find an angle, use
an inverse trig function.
• If sin = y/r then = sin-1 (y/r)
• Also, angles in the triangle add up to 90° + = 90°
• Complementary anglessin α = cos β
Using Trig Functions to Find Vector Components
Pythagorean Theorem
We can use all of this to
Add Vectors Analytically!
Components of Vectors• The x- and y-components
of a vector are its projections along the x- and y-axes
• Calculation of the x- and y-components involves trigonometry
Ax = A cos θAy = A sin θ
Vectors from Components• If we know the components,
we can find the vector.• Use the Pythagorean
Theorem for the magnitude:
• Use the tan-1 function to
find the direction:
ExampleV = Displacement = 500 m, 30º N of E
Example• Consider 2 vectors, V1 & V2. We want V = V1 + V2
• Note: The components of each vector are one-dimensional vectors, so they can be added arithmetically.
We want the 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.)
3. Add the components in each direction. (Vx = V1x + V2x, etc.)
4. Find the length & direction of V by using:
Adding Vectors Using Components•We want to add two vectors:
•To add the vectors, add their components
Cx = Ax + Bx Cy = Ay + By
• Knowing Cx & Cy, the magnitude and direction of C can be determined
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?
Solution, page 1 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, page 2 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?
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.
Solution, Page 1 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, Page 2 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.
Problem Solving
You cannot solve a vector problem without drawing a diagram!
Another Analytic Method • Uses Law of Sines & Law of Cosines from trig.• Consider an arbitrary triangle:
α
β
γ
a
b
c
• 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. Given A, B, γ
B
A
γ
β
α
AC
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
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