Method #2: Resolution into Components

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.

Quick Review

Right Triangle

a

c

b

A

B

C

c is the hypotenuse

c2 = a2 + b2

sin = opp/hyp cos = adj/hyp tan = opp/adj

A + B + C = 180°

transverse line crossing parallel lines: A A == A

AA + B = 90 °

AA

AA

Let’s look at one vector’s components:

To resolve a vector into perpendicular components

37o

100

Construct a line parallel to x through tailConstruct a line parallel to y through headArrows point the way from tail to head

37o

100

x

yUsing trig functions solve for x & y

X = 100cos 37o = 80Y = 100sin 37o = 60

Why is this important? Components of Force

x

y

Method #2: Adding Vectors By Resolution into Components USE COLOR PENCILS!USE COLOR PENCILS!

Stan is trying to rescue Kyle from drowning. Stan gets in a boat and travels at 6 m/s at 20o N of E, but there is a current of 4 m/s in the direction of 20o E of N. Find the velocity of the boat.

Don’t measure anything for this method!

Method #2: Adding Vectors By Resolution into Components USE COLOR PENCILS!USE COLOR PENCILS!

Stan is trying to rescue Kyle from drowning. Stan gets in a boat and travels at 6 m/s at 20o N of E, but there is a current of 4 m/s in the direction of 20o E of N. Find the velocity of the boat.

Don’t measure anything for this method!

Method #2: Adding Vectors By Resolution into Components USE COLOR PENCILS!USE COLOR PENCILS!

Stan is trying to rescue Kyle from drowning. Stan gets in a boat and travels at 6 m/s at 20o N of E, but there is a current of 4 m/s in the direction of 20o E of N. Find the velocity of the boat.

Don’t measure anything for this method!

Solve the following problem using the component method.

10 km at 30 N of E

6 km at 30 W of N

Solve the following problem using the component method.

10 km at 30 N of E

6 km at 30 W of N

Ry = Ay + By

Rx = Ax - Bx

Ay

Ax

By

Bx

R1. Solve for components using: SOH CAH TOA

2. Solve RESULTANT using: R2 = Rx

2 +Ry2

tan Ө = Rx/Ry

Another Example:

5 N at 30° N of E 6 N at 45°

x y cos 30° = x/5

5 cos 30° = 4.33sin 30° = y/5

5 sin 30° = 2.5

cos 45 ° = x/66 cos 45 ° = - 4.24

sin 45 ° = y/66 sin 45 ° = 4.24

0.09 6.74

R = (0.09)2 + (6.74)2 R = 6.74 N

tan = 6.74/0.09 = 89.2°

30°

45°

6

5

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.

And Another Example:

50

30

37o

x

y

50

parallel to x

37o

30

neither parallel to x or y

Continued…

90 – 37 = 53o

30

x

y

x

y

53o

30

X = 30 Cos 53o = 18

Y = 30 Sin 53o = 24

50 18

24

=68

24

37o

Neither Parallel nor Perpendicular Vector Addition (con)

68

24

For these perpendicular vectors

Find resultant magnitude & direction

68

24R

θR2 = 682 + 242

R = 72.1

tan θ = 24/68 = tan-1 24/68 = 19.4o N of E

This completes Method Two!

So lets keep

And practice some more! problems #3, 4 due tomorrow