1. Vectors and Scalars
AP Physics C

2. Scalar
A SCALAR is ANY quantity in physics that has MAGNITUDE, but NOT a direction associated with it.
Magnitude – A numerical value with units.
Scalar Example
Magnitude
Speed
20 m/s
Distance
10 m
Age
15 years
Heat
1000 calories

3. Vector
Vector
Magnitude & Direction
Velocity
20 m/s, N
Acceleration
10 m/s/s, E
Force
5 N, West
A VECTOR is ANY quantity in physics that has BOTH MAGNITUDE and DIRECTION. .

4. Polar Notation
Polar notation defines a vector by designating the vector’s magnitude |A| and angle θ relative to the +x axis. Using that notation the vector is written:
F=12<210
How do we write this vector?

5. Polar Notation
In this picture we have a force vector of 12 Newtons oriented along the -x axis. However, polar notation is relative to the + x axis. Therefore, it would be characterized by
Answer: F= 12<180
In this last picture we have 2 vectors. They are characterized by:
Answer: C=2<30
D=4<-50 or 4<310
F=12< and C=2<30 and D=4<-50 or 4<310

6. Scalar Multiplication
Multiplying a vector by a scalar will ONLY CHANGE its magnitude.
How can we change the
direction of a scalar?
Thus if A = 12 < 105, then what does 2A equal ?
Answer: 2A=24<105
Thus if A = 12 < 105, then
what does – A equal?
Answer: -A= 12<285
OR A = 12<-75
-1/2A

7. Unit Vector Notation
An effective and popular system used in engineering is called unit vector notation. It is used to denote vectors with an x-y Cartesian coordinate system.

8. Unit Vector Notation =3j = 4i
J = vector of magnitude “1” in the “y” direction
= 4i
i = vector of magnitude “1” in the “x” direction
The hypotenuse in Physics is called the RESULTANT or VECTOR SUM.
The LEGS of the triangle are called the COMPONENTS 3j
Vertical Component
NOTE: When drawing a right triangle that conveys some type of motion, you MUST draw your components HEAD TO TOE. 4i
Horizontal Component

9. Unit Vector Notation J= 2i + 4j K=2i -5j
How would you write vectors J and K in unit vector notation?
J= 2i + 4j
K=2i -5j
J=2i + 4j K=2i – 5j

10. Magnitude of a vector
The symbol |A| means the magnitude of vector A. The magnitude of a vector is equal to the square root of the sum of the squares of the components of the vector. |A|= √(Ax2 + Ay2 )

11. Non-Collinear Vectors
When 2 vectors are perpendicular, you must use the Pythagorean theorem.
|R|=√((95km)2 + (55km)2
R= km
A man walks 95 km, East then 55 km, north. Calculate his RESULTANT DISPLACEMENT.
55 km, N
109.8 km
95 km,E

12. BUT…..what about the VALUE of the angle???
Just putting North of East on the answer is NOT specific enough for the direction. We MUST find the VALUE of the angle.
Find the value of the angle.
Write the resultant vector in the following:
Polar Notation
Cartesian unit vector form
109.8 km
55 km, N q
N of E
95 km,E
R = km<30
or km, 30 degrees North of East
Or 95i + 55j
109.8 km< or km, 30 degrees N of E i +55j

13. What if you are missing a component?
Suppose a person walked 65 m, 25 degrees East of North. What were his horizontal and vertical components?
H.C. = ?
VC = 65cos25 VC= j HC=65sin25 HC 27.47i
V.C = ? 25
65 m
58.91 j i