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