Mathematical Investigations IV
Name:
Vectors
Getting To the Point
On the Map
Meet the vector!
Tip
(pointy end)
direction
Magnitude
(length)
Tail
(start)
These vectors are all equal:
The tail of a vector is not “glued” to the origin. A vector can originate anywhere! What makes them
equal is that they all have the same length and they point in the same direction.
The following notation is used to describe a vector mathematically:
The vector v  a , b  if often represented by an arrow. It is equivalent to the vector with its tail at
the origin and its head at the point (a, b) on the plane.
b
(a, b)
v
a
Vectors 1.1
Rev S07
Mathematical Investigations IV
Name:
We may write v (written with a little vector over it) or v (bold-faced, often used on the printed page).
For any vector, all of these notations may be used.
r
r a 
v   a ,b  , or v  aiˆ  bĵ, or v    .
b 
a is called the x-component of v
b is called the y-component of v
Note: iˆ is the name for the unit vector in the x-direction. It has length 1.
ĵ is the name for the unit vector in the y-direction. It also has length 1.
(more on this in a few days!)
Draw a few yourself!




t   1, 3 
u  2 i 2 j
3 
v   
4 
w   6,  4 
The two key attributes of a vector are its:
1. length or magnitude
2. direction (often given as the angle between the positive x-axis and the vector)
For the vector v   a , b  ,
the magnitude is given by
| v |  a 2  b2
Vectors 1.2
Rev S07
Mathematical Investigations IV
Name:
For each of the vectors t , u , v , and w on page 1.2, find
a) its magnitude
b) the angle between the vector and the positive x-axis
(Hint: Use right triangle trig.)
Note: When dealing with vector angles, we generally use degrees rather than radians.
vector
t
u
v
w
magnitude
|t| =
|u| =
|v| =
|w| =
angle (degrees)
As a concept, the vector leads a dual life.
1) Vectors have a rich mathematical life in the abstract realm. They exist in 2-D (as in these simple
examples), in 3-D (as we will see) and in n-dimensions (as we will also see). They play an
important part of linear algebra, vector calculus, and other fields.
2) Vectors are a convenient and elegant way to express certain physical properties. They are used to
describe such basic quantities as force, velocity, acceleration, electric and magnetic fields,... the
list continues. Vectors play a central role in the mathematics of physics and engineering.
Here is a prime example of a vector: velocity.
Velocity is defined as speed in a certain direction.
Consider the directions (N, S, W, E) on the coordinate plane:
N
W
E
S
Accordingly, we can express:
v = (75 m/s, South) = < 0, –75>
N
W
E
v = (35 m/s, West) = < –35, 0 >
N
W
35
E
75
S
S
Vectors 1.3
Rev S07
Mathematical Investigations IV
Name:
v = (50 km/h, NE) =
 25
2, 25 2 
N
50
W
E
S
Draw each of the following velocity vectors and write in terms of x- and y- components:
v = (60 km/h, N) = <
,
>
v = (80 m/s, SW) = <
v = (15 m/s, SE) = <
,
>
,
v = (55 km/h, 15 E of N)= <

(which may be written N15 E)
v = (25 km/h, N30W) = <
,
>
v = (70 m/s, S40E) = <
Vectors 1.4
,
>
,
>
>
Rev S07
Mathematical Investigations IV
Name:
When giving the angle direction, as in the last two cases, the first direction should be either N or S
(which is easily found with a compass), followed by a degree measure, followed by W or E
(i.e. either left or right of north or south).
In navigational circumstances, such as aviation, directions are given as bearings.
Instead of using the positive x-axis (East as we’ve drawn it) as the 0 reference, due North has a
bearing of 0. Bearing angles are measured clockwise from North. So, East has a bearing of 90,
South has a bearing of 180, and West has a bearing of 270.
N
W
E
S
Draw a few more:
v = (10 knots, Bearing of 45)
v = (150 mph, Bearing of 210)
Draw each of the following velocity vectors and write in terms of x- and y- components:
v = (20 km/h, Bearing of –60)
=<
,
>
v = (80 m/s, Bearing of 70)
=<
Vectors 1.5
,
>
Rev S07