Vectors 2 Components

The Right Triangle hypotenuse opposite θ adjacent

Pythagorean Theorem hypotenuse2 = opposite2 + adjacent2 c2 = a2 + b2 θ

Basic Trigonometry functions sin θ = opposite/hypotenuse cos θ = adjacent/hypotenuse tan θ = opposite/adjacent SOHCAHTOA hypotenuse opposite θ adjacent

SOHCAHTOA Inverse functions θ θ = sin-1(opposite/hypotenuse) θ = cos-1(adjacent/hypotenuse) θ = tan-1(opposite/adjacent) SOHCAHTOA hypotenuse opposite θ adjacent

Sample problem A surveyor stands on a riverbank directly across the river from a tree on the opposite bank. She then walks 100 m downstream, and determines that the angle from her new position to the tree on the opposite bank is 50o. How wide is the river, and how far is she from the tree in her new location?

Vectors: x-component A  Ax The x-component of a vector is the “shadow” it casts on the x-axis. cos θ = adjacent ∕ hypotenuse cos θ = Ax ∕ A Ax = A cos  y A  x Ax

Vectors: y-component The y-component of a vector is the “shadow” it casts on the y-axis. sin θ = opposite ∕ hypotenuse sin θ = Ay ∕ A Ay = A sin  y A Ay Ay  x

Vectors: angle The angle a vector makes with the x-axis can be determined by the components. It is calculated by the inverse tangent function  = tan-1 (Ry/Rx) y Ry  x Rx

R Vectors: magnitude Ry Rx The magnitude of a vector can be determined by the components. It is calculated using the Pythagorean Theorem. R2 = Rx2 + Ry2 y R Ry x Rx

Practice Problem Find the x- and y-components of the following vectors R = 175 meters @ 95o

Practice Problem Find the x- and y-components of the following vectors v = 25 m/s @ -78o

Practice Problem You are driving up a long inclined road. After 1.5 miles you notice that signs along the roadside indicate that your elevation has increased by 520 feet. What is the angle of the road above the horizontal?

Practice Problem You are driving up a long inclined road. After 1.5 miles you notice that signs along the roadside indicate that your elevation has increased by 520 feet. How far do you have to drive to gain an additional 150 feet of elevation?

Vector Addition by Component

Component Addition of Vectors Resolve each vector into its x- and y-components. Ax = Acos Ay = Asin Bx = Bcos By = Bsin Cx = Ccos Cy = Csin etc. Add the x-components (Ax, Bx, etc.) together to get Rx and the y-components (Ay, By, etc.) to get Ry.

Component Addition of Vectors Calculate the magnitude of the resultant with the Pythagorean Theorem (R = Rx2 + Ry2). Determine the angle with the equation  = tan-1 Ry/Rx.

Practice Problem In a daily prowl through the neighborhood, a cat makes a displacement of 120 m due north, followed by a displacement of 72 m due west. Find the magnitude and displacement required if the cat is to return home.

Practice Problem If the cat in the previous problem takes 45 minutes to complete the first displacement and 17 minutes to complete the second displacement, what is the magnitude and direction of its average velocity during this 62-minute period of time?