Vectors

The Basics: the notes How to add vectors Vectors should be added from “tip/head to tail” For example 4m west + 3m south would be 4 meters west 3 meters south

4 meters west 3 meters south resultant Tip of final arrow Starting point The answer or resultant will have both a magnitude (number) and a direction. The resultant (usually dashed) should be drawn from the starting point to the tip of the last arrow head ALWAYS INCLUDE AN ARROWHEAD ON YOUR RESULTANT! At this point, the magnitude and direction of the answer/resultant could be calculated…

Finding the resultant’s magnitude and direction http://phet.colorado.edu/sims/vector-addition/vector-addition_en.html

Finding the Magnitude (5 m) C = 5m C = A2 + B2 When using vectors there will always be triangles so Use the Pythagorean theorem A2 + B2 = C2 (where C is the hypotenuse) C = 5m C = 3m2 + 4m2 C = A2 + B2 4 m 3 m (5 m) We have found the magnitude of the vector Resultant=

Finding the direction Sin θ = opposite/hypotenuse Use trigonometry to find the angle at the base of the resultant vector Call this angle theta (θ) Sin θ = opposite/hypotenuse Cos θ = adjacent/hypotenuse Tan θ = opposite/adjacent Adjacent side θ 4 m Opposite side 3 m hypotenuse (5m) base

Finding the direction θ (5m) 4 m base Adjacent side Opposite side 3 m hypotenuse (5m) base

Finding the direction cont. When choosing which sides to use… choose the original givens not the side you calculated In this case the 3 meters and 4 meter side Which function should you use? Sin θ = opp/hyp Cos θ = adj/hyp Tan θ = opp/adj Adjacent side θ Tan θ = opp/adj 4 meters west Opposite side 3 meters south (5m)

Finding the direction cont. Tan θ = opp/adj Tan θ = 3m/4m You will need to use the inverse tan (2nd function- then push tan) on the cal. Cal= 36.86989765 R= θ = 400 Adjacent side 4 meters west θ 400 Opposite side 3 meters south 5m

Finding direction cont Direction will be given in degrees from 0 (due east) 900 1800 00= due east 2700

Place the resultant vector on a coordinate plane (the base will always be at the origin) 900 1800 00= due east Base of resultant vector 5 m 2700

5 m Final answer= 180 + 40 Final answer= 2200 We also know the angle we just calculated Now determine how far from 00 east this angle is… Final answer= 180 + 40 Final answer= 2200 We have now found the direction of the resultant vector! 900 1800 00= due east 400 Base of resultant vector 5 m 2700

Basic 2 Vector problem: [Two Forces act on an object. One is a northern force of 24.0 N. The other is an easterly force of 13.0 Newtons. What is the resultant force magnitude and direction on the object overall?]

*Final answer must be from 00 (due east) Resultant magnitude: (opp of θ) Resultant direction: 13.0 N 900 Tan θ = opp/adj Tan θ= 13.0N/24.0 N 24.0N Cal=28.44292862 (adj of θ) (27.3 N) clip..\.OB θ will be the angle at the base of the resultant 28.40 R= 28.40 θ ? 61.60 *Final answer must be from 00 (due east) Resultant magnitude: A2 + B2 = C2 C = A2 + B2 90.0 – 28.4= 61.60 C = 24.0N2 + 13.0N2 R= 27.3 N Cal = 27.29468813

One dog… three leashes However, vectors must be added head to tail This shows the reality of the situation…

2.5 N The order added doesn’t matter- the resultant is the same! 5.5 N 5.7-6.0 N Up and left And also… 1 leash is enough! 4.0 N

Rem., order does not matter

Bellwork: Date: Two students pull on a wishbone. One pulls at 34.9 N west, the other at 13.3 N south. 1) Draw the reality of the situation with labels. (not tip to tail- but how it would actually look from above) 2) What is the resultant magnitude? Must re-draw vectors tip to tail 3) What is the resultant direction? 37.3 N 200.90 (180.0000 + 20.9=) (one past decimal- NOT sig. figs.!)

Component Vectors http://phet.colorado.edu/sims/vector-addition/vector-addition_en.html

Vector Components Consider a 24 N force at 30.00 24 N 30.00

Vector Components To the right And upward What two directions does this arrow point? To the right And upward We say that this vector has both a Easterly component and a Northerly component 24 N 30.00

Vector Components cont. Calculate how much of the force of this vector acts in the northerly direction? Note that you have a side (24N) and an angle(30.00). Note also that the northern component is OPPOSITE the angle given. We also know the HYPOTENUSE (24N) Which function will be used? Sin θ = Opp/Hyp Sin 30.0 = opp/ 24N Cal= 12 R= 12 N northward 24 N (12N) 30.00

Vector Components cont. Calculate how much of the force of this vector acts in the Easterly direction.` Note also that the eastern component is ADJACENT to the angle given. HYPOTENUSE (24N) Which function will be used? Cos θ = Adj/Hyp Cos 30.0 = adj/ 24N Cal= 20.78460969 R= 21 N eastward 24 N 30.00 (21N) Vector components clip..\..\

Bellwork (pre-mini warm-up): Basic 2 vector and Component vectors This represents a boat and a current: A) 25.9 m/s east B) 33.3 m/s south 1) What is the resultant magnitude? 2) What is the resultant direction? 3) If a car is traveling 11.1 m/s at 280.00, how fast is it moving eastward? (Use the 10.0 not the 80.0 degree triangle) 42.2 m/s 52.10 360-52.1= 307.9 1.93 m/s east