3.2 Vector Operations

ReviewFind the x and y component.

Go 4 m to the right, 2 m up, 5 m down, 7m left, 3m to the left and 10 m up.

X= 6m left y = 7m up

Pythagorean’s Theorem

c2 = a2 + b2

c a

b

SOH CAH TOA

• Sin = opposite hypotenuse

• Cos = adjacent hypotenuse

• Tan = oppositeadjacent

Head to Tail Method

Helpful Hints to Solving Vector Problems

1. Place a dot on your paper

2. Draw the first vector given in the problem in the appropriate direction.

3. Without picking your pencil point off the paper draw the next vector in the appropriate direction.

4. Continue this with as many vectors as you have.

Helpful Hints to Solving Vector Problems

5. Draw your resultant from the dot in the beginning to the last vector arrow

6. Look at your drawing. You need to make a right triangle opposite your resultant.

7. Use pythagorean’s theorem to find the resultant side c

8. Vectors must have direction so you will use tan to find the angle from the resultant to the vector. (This angle always rotates around the original dot)

Direction

N

W E

S

A plane travels from Huston, Texas to Washington, DC, which is 1540 km, East and 1160km north of Houston. What is the total displacement of the plane?

C 1160 KM

1540 km

c2 = a2 + b2

c2 = (1540 km)2 + (1160km)2

C = 1928 km = 1930 km

Tan = 1160

1540

Answer: 1930 km, north of east Practice: 90 degree Problems

Boat Problems

A motorboat heads due east at 16 m /s across a river that flows due south at 9.0 m /s.

A. What Is the resultant velocity of the boat.

B. If the river is 136 m wide, how long does it take the motorboat to reach the other side?

C. How far downstream is the boat when it reaches the other side of the river?

What Is the resultant velocity of the

boat? 16 m /s

c 9 m /s

C 2 = (16)2 + (9)2 C = 18 m /s

Tan 9 / 16 = 30o

18 m /s, 30o south of east

If the river is 136 m wide, how long does it take the motorboat to reach

the other side?

V = d t

t = d = 136 m V 16 m /s

t = 8.50 s

How far downstream is the boat when it reaches the other side of the river?

V = d

t

d = V t

d = 9.0 m /s X 8.5 s = 77 m

Practice: Boat problems

Components

You are given the resultant (magnitude and direction) and you find the sides or components

Use sin and cos to find the x and y component

Hypotenuse angle y

x

Component Example

An arrow is shot from a bow at an angle of 25o above the horizontal with an initial speed of 45 m /s. Find the horizontal and vertical components of the arrow’s initial velocity.

45 m /s

25o y X

Components

Sin 25o = y = 19 m /s 45

Cos 25o = x = 41 m /s 45

Practice problems on components

Putting it all together

A mouse runs east for 4 m across the living room, turns 30o north of east and runs 2.8 m and finally heads due north 3 m to a hole in the wall. What is the resultant?

3 m

2.8 m 30o

4 m

Draw and examine. Look for a right triangle. The resultant is “from where you start to where you end it does not

matter where you have been”. Find the components to make the right triangle.

3m C

q 1.4 m

4m 2.4 m

• Add up the all the x’s then all the y’s: x total = 6.4m y total = 4.4m• Use pythagorean’s theorem c 2 = a2 + b2 C = 7.8 m• tan = 4.4 / 6.4 = 35 o

7.8 m, 35 o North of East

A camper walks 4.5 km at 45o north of east then 4.5 km due south. Find the

campers total displacement.

4.5

45o 4.5

4.5 sin 45 = x x = 3.184.5 sin 45 = y1= 3.18

4.5 - 3.18 = 1.32c2 = 3.182 + 1.322 c = 3.4 tan = 1.32 / 3.18 = 22o

3.4 m, 22o south of east

4.5 45o 4.5

Practice, Practice, Practice

• Practice #1

• Practice #2

• Holt Text Pg 94 1-4

• QUIZ