Vector

Components

Calculator Trig FunctionsMake sure calculator is in DEG

NOT RAD or GRAD (turn off/on)

Practice: find the followingcos 30ºsin 30ºcos 60ºsin 60ºcos 45ºsin 45ºcos 23ºsin 23º

0.8660.5000.500 0.8660.707 0.707 0.921 0.391

Vector ComponentsAny vector is the vector sum of an x-

component and a y-componenta = ax + ay

a

ax

ayy

x

θ

Vector ComponentsFind the magnitude of the vector components

ax = a cosθ ay = a sinθ

a

ax

ayy

x

θ

ExampleResolve a vector of 5 cm at 60º into it’s x and

y components:ay = a sinθ ax = a cosθ

= 5 sin 60º = 5 cos 60ºay = 4.33 ax = 2.5

note: sin 60º = 0.866 cos60º = 0.5

ExampleResolve 10 cm at 45º into x and y

ay = a sinθ ax = a cosθ

= 10 sin 45º = 10 cos 45ºay = 7.07 cm ax = 7.07 cm

10 cm at 45º has positive x and y components.

x

-y

-x

y

0º

270º

180º

90º

ExampleResolve a vector of 4 m/s at 143º

ay = a sinθ ax = a cosθ= 4 sin 143º = 4 cos 143º

ay = 2.41 m/s ax = -3.19 m/s

note: sin 143º = 0.602 cos 143º = -0.799 why negative?

4 m/s at 143º has a positive y but a negative x component

x

-y

-x

y

0º

270º

180º

90º

ExampleResolve a vector of 92 m/s2 at 230º

ay = a sinθ ax = a cosθ= 92 sin 230º = 92 cos 230º

ay = - 70.5 m/s ax = -59.1 m/s

note: sin 230º = -0.766 cos 230º = -0.642

92 m/s2 at 230º has a negative x and y component

x

-y

-x

y

0º

270º

180º

90º

ExampleYou walk northeast 4000 m. (That’s 45º

north of east). How far east and how far north have you walked?

N

W

S

E

4000 m

45º

East: = 4000 cos 45= 4000 x 0.707= 2828 m

North: = 4000 sin 45= 4000 x 0.707= 2828 m

2828m N

2828m E

4000 m

45º

Adding Vectors w/ Components1. Find x and y components of all vectors2. Add all x-components3. Add all y-components4. You now have rx and ry

5. Use Pythag. to find magnitude6. Use θ = tan-1(ry/rx) to find the angle

Example 1: Add a and ba = 3.0 N @ 20ºb = 2.0 N @ 45º

a

b+y

+x

Example: Add a and ba = 3.0 N @ 20º b = 2.0 N @ 45º

ax = 3 cos 20º bx = 2 cos 45º

= 2.82 N (x) = 1.41 N (x)ay = 3 sin 20º by = 2 sin 45º

= 1.03 N (y) = 1.41 N (y)

Add x and y componentsax + bx =2.82 N + 1.41 N

= 4.23 N (x) = rx

ay + by = 1.03 N + 1.41 N

= 2.44 N (y) = ry

Use Pythag. to find resultant rr2 = (rx)2 + (ry)2

r2 = (4.23)2 + (2.44)2

r2 = 23.85r = 4.88 N

rx = 4.23

ry = 2.44

r

θ

Use θ = tan-1(ry/rx) to find angleθ = tan-1(ry/rx)

= tan-1(2.44/4.23)= tan-1 (0.577)

θ = 30º

rx = 4.23

ry = 2.44r

θ

Final Result

a

b+y

+x

r = 4.88 @ 30º

b

30º

a = 4.0 m/s @ 35º b = 5.6 m/s @ 140º

ax = 3.28 m/s bx = -4.29 m/s

ay = 2.29 m/s by = 3.60 m/s

ax + bx = -1.01 m/s (x) = rx

ay + by = 5.89 m/s (y)= ry

ab +y

+x-xExample 2

r2 = (-1.01)2 + (5.89)2

r2 = 35.71r = 5.98 Nθ = tan-1(ry/rx)

= tan-1(5.89/-1.01)= tan-1 (-5.83)

θ = -80.3º ???θ = 180 + -80.3º = 99.7

rx = -1.01

ry = 5.89

r = 5.98

θ

Final Result

ab

+y

+x

ar

θ θ = 99.7