1. WHAT’S THE SHAPE? Y component Vector V Y X component Vector V X Original Vector Component – Means to be a piece, or apart, of something bigger A Component Vector is a smaller vector that is apart of a larger one LOOK AT THE VECTOR,AND THE AXES. WHAT SHAPE CAN YOU MAKE?

2. ANALYTICAL METHOD Use trigonometry to break vectors into smaller vectors called Component Vectors. –The X component Vector is along the X axis –The Y component Vector is along the Y axis Trigonometry deals with angles, and triangles. This makes addition easier. Please remember Trig. is your friend. We can add these component vectors to find the resultant.

3. TRIGONOMETRY SINE  = opposite/hypotenuse COS  = adjacent/hypotenuse TAN  = opposite/adjacent SOH - CAH - TOA Hypotenuse Adjacent Opposite 

4. BREAKING INTO COMPONENTS Vector (V) y component (V Y ) x component (V X )  Sine  = Y/V V Y = V*Sine  Cos  = X/V V X = V*Cos  Use Sine and Cosine: 1) Cosine closes the angle 2) Sine skies, Cosine coasts

5. BREAK INTO COMPONENTS First use sine, and cosine to break the vectors into their components Now that the original vector are broken into smaller pieces the original two vectors are gone

6. Adding the components Now, we can add these vectors easily by the tip tail method Resultant When we draw the resultant we get a right triangle, we get a right triangle EVERY TIME WE DO THIS.

7. FINDING THE RESULTANT Use Pythagorean Theorem to find the magnitude V R 2 = X T 2 + Y T 2 V R = X T 2 + Y T 2 To find the direction use the tangent Tan  R = Y T /X T  R = Tan -1 (Y T /X T ) Resultant vector (V R ) X Total (X T ) Y Total (Y T ) RR  Means Angle

8. Solving a problem First use sine, and cosine to break the vectors into their components 20 meters at 30 degrees + 10 meters at 60 degrees 20 m 30 O 60 O 10 m V x = 20m *Cos 30 V x = 20m *.866 V x = 17.32 m V Y = 20m *Sin 30 V Y = 20m *.5 V Y = 10 m V x = 10m *Cos 60 V x = 10m *.5 V x = 5 m V Y = 10m *Sin 60 V Y = 10m *.866 V Y = 8.66 m Now that the original vector are broken into smaller pieces the original two are gone

9. V x = 17.34 m V Y = 10 m V x = 5 m V Y = 8.66 m Now that we have these component vectors we can add them Tip Tail and form a right Triangle Once we have the right triangle we can find the lengths of each side by simply adding the vector length’s together V Y = 10 m V x = 5 m V Y = 8.66 m Resultant Now we have a regular right triangle and by using trigonometry we can find the resultant 10 m + 8.66 m = 18.66 m 17.32m + 5m = 22.34 m

10. 18.66 m 22.32 m C 2 = a 2 +b 2 C 2 = (22.32m) 2 + (18.66m) 2 C 2 = (498.18m 2 ) + (348.2m 2 ) C 2 = 846.38m 2 C = 846.38m 2 = 29.09m a = 22.34m b = 18.66 m C= resultant Resultant  = tan -1 (b/a)  = tan -1 (18.66m/22.32m )  = tan -1 (.836)  = 39.9 O  The final answer is 29.09 meters at 39.9 O

11. WOW, That a lot of work!! (Is there an easier way, to do this?) Yes, there is a much faster and easier way, called the “box method” The box method is a technique that organizes all the work into a few simple steps that anyone can do Lets look at the same problem again using the box method, on the side will be the longer method In the end chose the method that works best for you.

12. The box method Step 1: Make a box like this one the top, and bottom rows are always done this way Vectors V x = Vcos (  )V y = Vsin (  ) Total You will need 1 row for each Vector. This problem has 2 vectors so we need 2 columns

13. Step 2: Fill in the vector columns (this sets up the problem) Vectors V x = Vcos (  )V y = Vsin (  ) 20 m at 30 O 10 m at 60 O Total 20 meters at 30 degrees + 10 meters at 60 degrees Write the Vector’s Length and Angle In the Vector column 20m*cos(30 O ) 17.32 m 20m*sin(30 O ) 10 m 10m*cos(60 O ) 5 m 10m*sin(60 O ) 8.66 m Then use The Cosine And sine Equations To fill in the next two columns Finally add up the two Columns 22.32 m 18.66 m

14. Using the total columns gets us the right triangle again 18.66 m 22.32 m C 2 = a 2 +b 2 C 2 = (22.32m) 2 + (18.66m) 2 C 2 = (498.18m 2 ) + (348.2m 2 ) C 2 = 846.38m 2 C = 846.38m 2 = 29.09m V x total = a = 22.34m V y total = b = 18.66 m C= resultant Resultant  = tan -1 (b/a)  = tan -1 (18.66m/22.32m )  = tan -1 (.836)  = 39.9 O  The final answer is 29.09 meters at 39.9 O