1. Vector Components & Adding Non-perpendicular Vectors
Sept 1, 2016

2. Vector Components
Any vector can be “resolved” into two component vectors.
How do we calculate Ax and Ay ?
Use trig!
cos q = 𝐴𝑥 𝐴 sin q = 𝐴𝑦 𝐴
Ax = A cos q Ay = Asin q A Ay q Ax
Ax is the horizontal component – or x component -- of the vector.
Ay is the vertical component – or the y component – of the vector.

3. vy vx v = 34 m/s @ 48° . Find vx and vy
Example: A plane heads east, while the wind moves a plane north. As a result, the plane moves with velocity of 34 48°relative to the ground.
Calculate the plane's heading and wind velocity.
What does this mean??
It means we need to find the x-component of the plane’s resulting velocity (= wind velocity) and the y-component of the plane’s resulting velocity (= plane’s heading).
v = 34 48° . Find vx and vy
cos 48o = 𝑉𝑥 34 𝑚/𝑠
vx = 34 m/s cos 48° = 23 m/s
sin 48o = 𝑉𝑦 34 𝑚/𝑠
vy = 34 m/s sin 48° = 25 m/s vy q vx

4. A plane moves with a velocity of 63.5 m/s at 32 degrees South of East.
Calculate the plane's horizontal and vertical velocity components.
cos (32 0 ) = 𝑣 𝑥 𝑚/𝑠
vx = ?
𝑣 𝑥 =63.5 cos (32 0 ) =53.9 𝑚 𝑠 𝐸
320
Vy = ?
63.5 m/s
sin (32 0 ) = 𝑣 𝑦 𝑚/𝑠
𝑣 𝑦 =63.5 sin ( 32 0 ) =33.6 𝑚 𝑠 𝑆

5. Problems for you to try individually
A person walks degrees. Find the x and y component vectors.
A car accelerates 6 m/s2 at 40 degrees. Find the x and y component vectors.

6. Problems for you to try individually
A person walks degrees. Find the x and y component vectors.
-225 m = Ax
390 m north = Ay
A car accelerates 6 m/s2 at 40 degrees. Find the x and y component vectors.
5 m = Ax
4 m = Ay

7. You can find a vector from its components.
This problem may be written differently, but its exactly the same type of problem we did during our first lesson on vectors!
Just add the components to find the overall vector!
Let:
Fx = 4 N
Fy = 3 N .
Find magnitude and direction of the vector.
Fx2+ Fy2 = F2 Fy q Fx
 = arc tan (¾) = 370

8. So far, we can …
Add two (or more) vectors that occur perpendicularly or in a line (day 1 of vectors)
Find the x and y components of a vector (today)
Determine a vector from its components (today)
Can we add two (or more) vectors if they occur at non-right angles?
Now, with a little graphical reasoning and vector components … we can!

9. Adding vectors at any angle
We start like we always do … moving vectors head to tail.
Next, we draw a resultant (as always).
3) Next step is also same as before … we use the other sides of a right triangle to calculate the magnitude and direction of the resultant (C).
The difference? We just have to do a little more reasoning to ‘see’ the right triangle and find its sides.
We can always – ALWAYS – find a right triangle that is made from combining the vector components of the vectors we are adding.
The resultant’s x component is equal to the sum of the x components of the vectors we are adding.
Cx = Ax + Bx
The resultant’s y component is equal to the sum of the y components of the vectors we are adding.
Cy = Ay + By
And we can figure all these out with trig! By Cy Bx Ay Ax Cx

10. Example: = 68 N@ 24° = 32 N @ 65° Solving procedure:
Two people are lugging a heavy suitcase. One pulls with a 68N force at 24o; the other pulls with a 32 N force at 65O. What is the total force on the suitcase?
= 68 24°
= 32 65°
Solving procedure:
1) Move vectors head to tail.
2) Draw resultant.
Draw the x and y components of each vector.
Find Fx by adding F1x + F2x
Find Fy by adding F1y + F2y
Calculate F using Pythagorean theorem
Calculate q by using tan-1

11. Example: Fx = F1x + F2x = 68 cos240 + 32 cos650 = 75.6 N
Two people are lugging a heavy suitcase. One pulls with a 68N force at 24o; the other pulls with a 32 N force at 65O. What is the total force on the suitcase?
= 68 24°
= 32 65°
Fx = F1x + F2x = 68 cos cos650 = 75.6 N
Fy = F1y + F2y = 68 sin sin650 = 56.7 N
 = arc tan (56.7/75.6) = 36.90

12. You try these individually
V1 = degrees
V2 = degrees
Find V = V1 + V2
X1 = degrees
X2 = degrees
Find X = X1 + X2
HINT: If x or y components point in opposite directions, then subtract the smaller from the larger!

13. Closure, HW, Exit Ticket Closure: How did what we do today …
… relate to our unit statement?
… demonstrate LP traits?
… relate to TOK?
HW: See handout.
HW Quiz next class, test in 2 classes!
Exit ticket: See handout