3. Vectors
We can use "+" and "-" to indicate the direction of one-dimensional motion. For motion in 3D, we use the concept of a vector.
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A vector has both a magnitude and a direction, represented graphically by an arrow with length proportional to magnitude.
e.g. displacement, velocity, and acceleration.
A scalar has a magnitude but no direction.
e.g. time and speed.
Displacement Vector:
Consider a car travelling north from OSU for
3 miles which then turns west for 2 miles and then south for 1 mile. What is the car's displacement from OSU?

2 miles
1 mile
3 miles
N r displacement vector
• displacement vectors describe the
change in position from start to finish, independent of the path taken.
Addition and Subtraction of Vectors:
Adding two vectors graphically by placing the end of one arrow to the tip of the other:
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A B
A B
C C = A + B
Treat subtraction as addition of a vector with a negative sign:

C
B
B
F = A B
A
B
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We can also add or subtract a series of vectors. Consider the car trip around OSU:
C B A
N
B
C
A r
Vector Components:

We can also add vectors by adding the individual components. To calculate the individual components, choose a Cartesian coordinate system: y
4
A y
A
θ
0
A x
A y
=
A cos
θ
=
A sin
θ
A x x
Addition of Vectors by Components:
C
= r
A
+
B
C
C x y
=
A x
=
A y
+
B
+
B y x

y
A A y
A x
θ
C
C x r
C
=
C x
2 +
C y
2
θ = tan
−
1
C y
C x
B x
B
B y
C y x
Subtraction of Vectors by Components:
F
= r
A
−
B
F
F x y
=
A x
=
A y
−
B
−
B y x
Similarly, in 3D:
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r
C
=
C
C x y r
A
+
=
A x
=
A y r
B
+
B x
+
B y
C z
=
A z
+
B z r
C
=
C
=
C x
2 +
C y
2 +
C z
2
Unit Vectors:
Vector with a length of 1.
y j
C
C y k i x
C z z
C x
Any vector can be written as a sum of unit vectors:
C
=
C e.g.
ˆ +
C y
D
=
10ˆ i
ˆ
−
+
C j z
+
3 ˆ
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y
+10
-5 x z
+3 asdf
Example:
A P131 student jogs east for 14 m, then turns south by 30° and runs 25 m, and then turns directly west and runs 15 m. What is the student's displacement?
y
14 m
A
D
30 x
E
B 25 m
S
C
15 m
7

8 r
A
=
A x
ˆ +
A y
ˆ = i r
B
=
B x r
C
=
C x
ˆ +
B y
ˆ +
C y
ˆ =
25cos30 o m ˆ
−
25sin 30 o
ˆ = − i
D
D x y
=
A x
=
A y
+
B
+
B y x
+
C
+
C y x
=
=
14 m
0 m
−
+
22 m
12.5 m
−
+
15 m
0 m
=
= −
21 m
12.5 m
D = D
2 x
+
D
2 y
= ( ) 2 (
12.5
) 2 =
24 m
θ = tan
−
1
D y
D x
= tan
−
1
 −
12.5

21

 =
31 o
(south of east)