Vector Components

Calculator Trig Functions Make sure calculator is in DEG NOT RAD or GRAD (turn off/on)

Practice: find the following cos 30º sin 30º cos 60º sin 60º cos 45º sin 45º cos 23º sin 23º

Vector Components Any vector is the vector sum of an x-component and a y-component a = a x + a y a axax ayay y x θ

Vector Components Find the magnitude of the vector components a x = a cosθa y = a sinθ a axax ayay y x θ

Example Resolve a vector of 5 cm at 60º into it’s x and y components: a y = a sinθ a x = a cosθ = 5 sin 60º= 5 cos 60º a y = 4.33 a x = 2.5 note: sin 60º = 0.866cos60º = 0.5

Example Resolve 10 cm at 45º into x and y a y = a sinθ a x = a cosθ = 10 sin 45º= 10 cos 45º a y = 7.07 cm a x = 7.07 cm

10 cm at 45º has positive x and y components. x -y -x y 0º0º 270º 180º 90º

Example Resolve a vector of 4 m/s at 143º a y = a sinθ a x = a cosθ = 4 sin 143º= 4 cos 143º a y = 2.41 m/s a x = m/s note: sin 143º = cos 143º = why negative?

4 m/s at 143º has a positive y but a negative x component x -y -x y 0º0º 270º 180º 90º

Example Resolve a vector of 92 m/s 2 at 230º a y = a sinθ a x = a cosθ = 92 sin 230º= 92 cos 230º a y = m/s a x = m/s note: sin 230º = cos 230º =

92 m/s 2 at 230º has a negative x and y component x -y -x y 0º0º 270º 180º 90º

Example You 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 = 2828 m North: = 4000 sin 45 = 4000 x = 2828 m 2828m N 2828m E 4000 m 45º

Adding Vectors w/ Components 1. Find x and y components of all vectors 2. Add all x-components 3. Add all y-components 4. You now have r x and r y 5. Use Pythag. to find magnitude 6. Use θ = tan -1 (r y /r x ) to find the angle

Example 1: Add a and b a = º b = º a b +y +x

Example: Add a and b a = ºb = º a x = 3 cos 20ºb x = 2 cos 45º = 2.82 N (x)= 1.41 N (x) a y = 3 sin 20ºb y = 2 sin 45º = 1.03 N (y)= 1.41 N (y)

Add x and y components a x + b x =2.82 N N = 4.23 N (x)= r x a y + b y = 1.03 N N = 2.44 N (y)= r y

Use Pythag. to find resultant r r 2 = (r x ) 2 + (r y ) 2 r 2 = (4.23) 2 + (2.44) 2 r 2 = r = 4.88 N r x = 4.23 r y = 2.44 r θ

Use θ = tan -1 (r y /r x ) to find angle θ = tan -1 (r y /r x ) = tan -1 (2.44/4.23) = tan -1 (0.577) θ = 30º r x = 4.23 r y = 2.44 r θ

Final Result a b +y +x r = 30º b 30º

a = º b = º a x = 3.28 m/s b x = m/s a y = 2.29 m/s b y = 3.60 m/s a x + b x = m/s (x)= r x a y + b y = 5.89 m/s (y)= r y a b +y +x -x Example 2

r 2 = (-1.01) 2 + (5.89) 2 r 2 = r = 5.98 N θ = tan -1 (r y /r x ) = tan -1 (5.89/-1.01) = tan -1 (-5.83) θ = -80.3º ??? θ = º = 99.7 r x = r y = 5.89 r = 5.98 θ

Final Result a b +y +x a r θ θ = 99.7