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 of COMPONENTS V x, V y Finding THE COMPONENTS V x & V y 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 V x, V y Finding V x & V y is EQUIVALENT to finding 2 mutually perpendicular vectors which, when added (with vector addition) will give V. That is, we want to find V x & V y such that V  V x + V y (V x || x axis, V y || 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 x- and y-axes

V is Resolved Into Components: V x & V y V  V x + V y (V x || x axis, V y || y axis) By the parallelogram method, clearly THE VECTOR SUM IS: V = V x + V y In 3 dimensions, we also need a V z.

Brief Trig Review Hypotenuse  h, Adjacent side  a Opposite side  o h o 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: r 2 = x 2 + y 2 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 angles sin 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 A x = A cos θ A y = 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:

Example V = Displacement = 500 m, 30º N of E

Example Consider 2 vectors, V 1 & V 2. We want V = V 1 + V 2 Note: The components of each vector are one- dimensional vectors, so they can be added arithmetically.

We want the sum V = V 1 + V 2 “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 V 1x, V 1y, V 2x, V 2y. (V 1x = V 1 cos θ 1, etc.) 3. Add the components in each direction. (V x = V 1x + V 2x, 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 C x = A x + B x C y = A y + B y Knowing C x & C y, 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: c 2 = a 2 + b 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 A C B Law of Cosines: C 2 = A 2 + B 2 -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