1. Method #2: Resolution into Components

2. Solving Vector Problems using the Component Method  Each vector is replaced by two perpendicular vectors called components.  Turn every vector into a right triangle.  Add the x-components and the y- components to find the x- and y- components of the resultant.  Use the Pythagorean theorem and the tangent function to find the magnitude and direction of the resultant.

3. Quick Review Right Triangle a c b A B C c is the hypotenuse c 2 = a 2 + b 2 sin = opp/hyp cos = adj/hyp tan = opp/adj A + B + C = 180° transverse line crossing parallel lines: A = A = A A A + B = 90 ° A A

4. Let’s look at one vector’s components: To resolve a vector into perpendicular components 37 o 100 Construct a line parallel to x through tail Construct a line parallel to y through head Arrows point the way from tail to head 37 o 100 x y Using trig functions solve for x & y X = 100cos 37 o = 80 Y = 100sin 37 o = 60

5. Why is this important? Components of Force x y

6. USE COLOR PENCILS! Method #2: Adding Vectors By Resolution into Components USE COLOR PENCILS! Stan is trying to rescue Kyle from drowning. Stan gets in a boat and travels at 6 m/s at 20 o N of E, but there is a current of 4 m/s in the direction of 20 o E of N. Find the velocity of the boat. Don’t measure anything for this method!

7. USE COLOR PENCILS! Method #2: Adding Vectors By Resolution into Components USE COLOR PENCILS! Stan is trying to rescue Kyle from drowning. Stan gets in a boat and travels at 6 m/s at 20 o N of E, but there is a current of 4 m/s in the direction of 20 o E of N. Find the velocity of the boat. Don’t measure anything for this method!

8. USE COLOR PENCILS! Method #2: Adding Vectors By Resolution into Components USE COLOR PENCILS! Stan is trying to rescue Kyle from drowning. Stan gets in a boat and travels at 6 m/s at 20 o N of E, but there is a current of 4 m/s in the direction of 20 o E of N. Find the velocity of the boat. Don’t measure anything for this method!

9. Solve the following problem using the component method. 10 km at 30 N of E 6 km at 30 W of N

10. Solve the following problem using the component method. 10 km at 30 N of E 6 km at 30 W of N R y = A y + B y R x = A x - B x AyAy AxAx ByBy BxBx R 1. Solve for components using: SOH CAH TOA 2. Solve RESULTANT using: R 2 = R x 2 +R y 2 tan Ө = R x /R y

11. Another Example: 5 N at 30° N of E 6 N at 45° xy cos 30° = x/5 5 cos 30° = 4.33 sin 30° = y/5 5 sin 30° = 2.5 cos 45 ° = x/6 6 cos 45 ° = - 4.24 sin 45 ° = y/6 6 sin 45 ° = 4.24 0.096.74 R =  (0.09) 2 + (6.74) 2 R = 6.74 N tan  = 6.74/0.09  = 89.2° 30° 45° 6 5

12. Advantages of the Component Method:  Can be used for any number of vectors.  All vectors are added at one time.  Only a limited number of mathematical equations must be used.  Least time consuming method for multiple vectors.

13. And Another Example: 50 30 37 o x y 50 parallel to x 37 o 30 neither parallel to x or y

14. Continued… 90 – 37 = 53 o 30 x y x y 53 o 30 X = 30 Cos 53 o = 18 Y = 30 Sin 53 o = 24 50 18 24 = 68 24 37 o

15. Neither Parallel nor Perpendicular Vector Addition (con) 68 24 For these perpendicular vectors Find resultant magnitude & direction 68 24 R θ R 2 = 68 2 + 24 2 R = 72.1 tan θ = 24/68 = tan -1 24/68 = 19.4 o N of E

16. This completes Method Two! So lets keep And practice some more! problems #3, 4 due tomorrow