Vector foundation for my AP class

1. Vector vs. Scalar
Vector Ex. Force
 magnitude & direction Weight
(amount)
Velocity
Scalar Acceleration
 magnitude Ex. Mass
Time
Money
Distance

2. Vector logistics
Vectors are represented by an arrow
Tells
direction
Length tells magnitude
Coordinate axis are used to represent position
North 10 N @ 45
+y 90°
180° +x
West -x East
0°
270° -y
South
Vectors are the same no matter where they are located, as long
as the magnitude and direction are the same!

3. Resultant
 a single vector that replaces two or more vectors
1. Adding vectors A = 5.00 N @ 0.00°
5 B = 7.00 N @ 0.00°
7 R = 12.0 N @ 0.00°
2. Subtracting vectors A = 5.00 N @ 0.00°
(special case of addition) B = 7.00 N @ 180°
5
7
A + B = 5 + -7 = -2 or R = 2.00 N @ 180°

5. 3. Adding at 90 A = 7.00 N @ 0.00°
B = 5.00 N @ 90.0°
Remember:
Resultant is from
beginning to end! 5
7
How do you draw this?
Parallelogram.
Draw dotted lines parallel to 5
original vectors.
Draw resultant from solid 7
lines to dotted lines.

6. How do you calculate the resultant?
Magnitude 
Pythagorean theorem
5
Direction 
SOHCAHTOA
2 2 opp 7
R 5 7 8.60N tan
adj
theta
Is the angle in  first
5
reference to the tan 1
inside
origin? If not, add 7 angle you
back to 0. 35.5 come to
from 0

7. yt. M = 54.0 N @ 10.0°
O = 35.0 N @ 100° R
35
2 2 54
R 54 35 64.4N Steps:
opp
tan 1.
adj
35 2. draw a picture
1
tan
54 3. pythagorean theorem
32.9 +10 = 42.9° 4. inverse tangent
5. add angle back to zero

8. Components
Components  2 vectors that replace 1 vector
1 is always on the x – axis M My
1 is always on the y – axis
A Mx
Ay yt Px
Cx
Py P
Ax C Cy
Components are always at right angles.
Components are independent of each other!

9. A = 87.0 m/s @ 40.0
How do you mathematically find Ay & Ax?
opp
SOHCAHTOA cos
adj sin
adj
hyp
Set angle theta. cos 40
Ax sin(40)
Ay
87 87
Ax 87 cos 40 Ay 87 sin(40)
A
Ay When finding components, it
will always be this way!
Ax = mag cos (angle)
Ax
Ay = mag sin (angle)
y sin, because x is cos

11. Vectors: 90°
To add vectors that are not at 90° to each
other, the vectors must first be broken
into their components.
ex. A = 10.4 m/s @ 75°
B = 6.7 m/s @ 25° R
Ay
A
B
By
Ax
Bx

12. Steps:
1. components  find xtot & ytot
2. draw a picture
3. pythagorean theorem
4. inverse tangent
5. add angle back to zero

13. ex. A = 10.4 m/s @ 75°
B = 6.7 m/s @ 25°
Ax = 10.4 cos 75 = 2.69 Ay = 10.4 sin 75 = 10.0
Bx = 6.7 cos 25 = 6.07 By = 6.7 sin 25 = 2.83
xtot = 8.76 ytot = 12.83
ytot = 12.83
R= (8.762 + 12.832)
R R = 15.5
θ = tan-1 (12.83/8.76)
θ = 55.9°
θ
R = 15.5 m/s at 55.9°
Xtot = 8.76