Warm up What is a vector? What is a force? How do we use vectors when analyzing forces?

Forces in two dimensions: Last chapter, we looked at forces in one dimension. Example: You push on a box with a force of 10N east and your friend pushes on the same box with a force of 5N east, what is Fnet? You push with 10N east and friend pushes 5N west, what is Fnet? Now we will look at forces in more than one direction. You push 10N east and friend pushes 10N north. What is Fnet? We use vector addition and triangles. Copyright © McGraw-Hill Education Vectors

Vocabulary Review New vector components vector resolution Vectors Copyright © McGraw-Hill Education

Vectors in Two Dimensions You can add vectors by placing them tip-to-tail. When you move a vector, do not change its length or direction. The resultant of the vector is drawn from the tail of the first vector to the tip of the second vector. Copyright © McGraw-Hill Education Vectors

Vectors in Two Dimensions If vectors A and B are perpendicular, use the Pythagorean theorem to find the magnitude of the resultant vector. Pythagorean Theorem R B A Copyright © McGraw-Hill Education Vectors

Example If you walk 10meters east and then 10 meters north, what is your displacement? If you push on a box with a force of 10N east and you friend pushes on a box with a force of 10N north, what is the net force? If the vectors are not at right angle to each other, we CANNOT use Pythagorean theorem. If we only know 1 side of the triangle we can not use Pythagorean. We need to use Trigonometry.

Trigonometry Notes

The Trigonometric Functions we will be looking at SINE COSINE TANGENT

The Trigonometric Functions SINE COSINE TANGENT

SINE Pronounced “sign”

COSINE Pronounced “co-sign”

Pronounced “tan-gent”

Represents an unknown angle Greek Letter q Pronounced “theta” Represents an unknown angle

hypotenuse hypotenuse opposite opposite adjacent adjacent

We need a way to remember all of these ratios…

Some Old Horse Came A Hoppin’ Through Our Alley

Sin SOHCAHTOA Opp Hyp Cos Adj Hyp Tan Opp Adj

Finding sin, cos, and tan. (Just writing a ratio or decimal.)

10.8 9 A 6 Find the sine, the cosine, and the tangent of angle A. Give a fraction and decimal answer (round to 4 places). 10.8 9 A 6 Shrink yourself down and stand where the angle is. Now, figure out your ratios.

24.5 8.2 23.1 Find the sine, the cosine, and the tangent of angle A Give a fraction and decimal answer (round to 4 decimal places). 8.2 A 23.1 Shrink yourself down and stand where the angle is. Now, figure out your ratios.

11/2/15 Hand in your article review Hand in your Newton’s laws handouts Take out your notebook so we can get reorganized.

Finding a side. (Figuring out which ratio to use and getting to use a trig button.)

Ex: 1. Figure out which ratio to use. Find x Ex: 1 Figure out which ratio to use. Find x. Round to the nearest tenth. 20 m x Shrink yourself down and stand where the angle is. tan 55 20 = Now, figure out which trig ratio you have and set up the problem.

Ex: 2 Find the missing side. Round to the nearest tenth. 80 ft x 3.1 72 tan = 80 ÷ 3.1 = Shrink yourself down and stand where the angle is. Now, figure out which trig ratio you have and set up the problem.

Ex: 3 Find the missing side. Round to the nearest tenth. Shrink yourself down and stand where the angle is. Now, figure out which trig ratio you have and set up the problem.

Ex: 4 Find the missing side. Round to the nearest tenth. 20 ft x

Finding an angle. (Figuring out which ratio to use and getting to use the 2nd button and one of the trig buttons.)

Use inverse function to solve for angle sinθ=opposite hypotenuse If: opposite=15m hypotenuse=30m sinθ=15m= 0.5 30m θ=sin-1 θ=0.5(sin-1)=30° θ hypotenuse adjacent opposite

Ex. 1: Find . Round to four decimal places. 17.2 9 tan-1 17.2 9 Shrink yourself down and stand where the angle is. Now, figure out which trig ratio you have and set up the problem. Make sure you are in degree mode (not radians).

Ex. 2: Find . Round to three decimal places. 7 2nd cos 7 23 ) 23 Make sure you are in degree mode (not radians).

Ex. 3: Find . Round to three decimal places. 200 400 2nd sin 200 400 ) Make sure you are in degree mode (not radians).

When we are trying to find a side we use sin, cos, or tan. When we are trying to find an angle we use sin-1, cos-1, or tan-1.

Vectors in Two Dimensions If the angle between the vectors is not 90°, you can use the law of sines or the law of cosines. Law of sines Law of cosines Copyright © McGraw-Hill Education Vectors

Vectors in Two Dimensions KNOWN UNKNOWN A = 20.0 N B = 7.0 N R = ? θ = 180.0° − 30.0° = 150.0° θR = ? Use with Example Problem 1. Problem Find the magnitude of the sum of two forces, one 20.0 N and the other 7.0 N, when the angle between them is 30.0°. Response SKETCH AND ANALYZE THE PROBLEM Draw a vector diagram and add the vectors graphically. List the knowns and unknowns. Draw the resultant vector. Click to continue. Place vectors tip to tail. Click to continue. Draw the initial vectors. Click to continue. ? 7.0 N 30.0° 20.0 N θ = 150.0° Copyright © McGraw-Hill Education Vectors

Vectors in Two Dimensions KNOWN UNKNOWN A = 20.0 N B = 7.0 N R = ? θ = 180.0° − 30.0° = 150.0° θR = ? Use with Example Problem 1. Problem Find the magnitude of the sum of two forces, one 20.0 N and the other 7.0 N, when the angle between them is 30.0°. SOLVE FOR THE UNKNOWN Use the law of cosines. Response SKETCH AND ANALYZE THE PROBLEM Draw a vector diagram and add the vectors graphically. List the knowns and unknowns. EVALUATE THE ANSWER This answer is consistent with this problem’s vector diagram, which shows that the resultant should indeed be slightly greater in magnitude than the 20.0-N force. ? 7.0 N 20.0 N θ = 150.0° Copyright © McGraw-Hill Education Vectors

Vector Components Concepts in Motion Vectors Copyright © McGraw-Hill Education Vectors

Vector Components A vector can be broken into its components, which are a vector parallel to the x-axis and another parallel to the y-axis. The process of breaking a vector into its components is sometimes called a vector resolution. Copyright © McGraw-Hill Education Vectors

Vector Components Complete the diagram using the terms positive and negative. Click on a question mark to reveal the answer. ? ? ? ? ? ? ? ? ? ? ? ? Copyright © McGraw-Hill Education Vectors

Algebraic Addition of Vectors Two or more vectors (A, B, C, etc.) may be added by first resolving each vector into its x- and y-components. +y +x Cy C B By A Ay Ax Bx Cx Copyright © McGraw-Hill Education Vectors

Algebraic Addition of Vectors Two or more vectors (A, B, C, etc.) may be added by first resolving each vector into its x- and y-components. The x- and y-components are added to form the x- and y-components of the resultant: +y +x By Ay Cy C B Ry Rx = Ax + Bx + Cx A Ry = Ay + By + Cy Bx Ax Cx Rx Copyright © McGraw-Hill Education Vectors

Algebraic Addition of Vectors Use the Pythagorean theorem to find the magnitude of the resultant vector: R2 = Rx2 + Ry2. Use trigonometry to find the angle of the resultant vector: +y +x Ry A B C R Rx Copyright © McGraw-Hill Education Vectors

Algebraic Addition of Vectors +x (East) +y (North) Algebraic Addition of Vectors θB = 135° B = 7.3 m Use with Example Problem 2. Problem Add the following two vectors via the component method: A is 4.0 m south B is 7.3 m northwest By Response SKETCH AND ANALYZE THE PROBLEM Establish a coordinate system. Draw a vector diagram. Include the knowns and unknowns in your diagram. Sketch the x-components and y-components of A and B. Bx θA = 270° A = 4.0 m Ay Copyright © McGraw-Hill Education Vectors

Algebraic Addition of Vectors +x (East) +y (North) Algebraic Addition of Vectors B = 7.3 m Use with Example Problem 2. Problem Add the following two vectors via the component method: A is 4.0 m south B is 7.3 m northwest By Response SKETCH AND ANALYZE THE PROBLEM Use your vector diagram as needed. Bx Rx = −5.16 m SOLVE FOR THE UNKNOWN Find the x-components of A and B and add them to find the x-component of R. Draw and label Rx on the vector diagram. A = 4.0 m Ay Copyright © McGraw-Hill Education Vectors

Algebraic Addition of Vectors +x (East) +y (North) Algebraic Addition of Vectors B = 7.3 m Use with Example Problem 2. Problem Add the following two vectors via the component method: A is 4.0 m south B is 7.3 m northwest By Ry = 1.16 m Response SKETCH AND ANALYZE THE PROBLEM Use your vector diagram as needed. Bx Rx = −5.16 m SOLVE FOR THE UNKNOWN Find the y-components of A and B and add them to find the y-component of R. Draw and label Ry on the vector diagram. A = 4.0 m Ay Copyright © McGraw-Hill Education Vectors

Algebraic Addition of Vectors +x (East) +y (North) Algebraic Addition of Vectors B = 7.3 m Use with Example Problem 2. Problem Add the following two vectors via the component method: A is 4.0 m south B is 7.3 m northwest R = 5.3 m Response SKETCH AND ANALYZE THE PROBLEM Use your vector diagram as needed. θR = 167° SOLVE FOR THE UNKNOWN R = 5.3 m at 13° north of west A = 4.0 m EVALUATE THE ANSWER Check the answer graphically. Vectors Copyright © McGraw-Hill Education

Review Essential Questions Vocabulary How are vectors added graphically? What are the components of a vector? How are vectors added algebraically? Vocabulary components vector resolution Copyright © McGraw-Hill Education Vectors