Mathematical Investigations IV
Name:
Vectors
Getting To the Point
Vector Arithmetic
There are four basic operations that we perform with vectors: addition, scalar multiplication, dot
(or inner) product, and cross (or vector) product. Let’s look at the first two operations.
How do we know how to steer an airplane in the wind? Or how to navigate a boat when there are
currents in the water? How do we calculate the load-bearing requirements of a bridge? Or of an
artificial knee?
Although these scenarios are all quite different from one another, the mathematics used in each
case is the same: vector addition.
Recall that vectors have magnitude and direction. How do we add two “arrows” together?
Here are two vectors, v1 and v2
v1
v2
Here are 3 ways to add them graphically, v1  v2  v3
v2
v3

v1
v1
v3

v1
v3
v2
v2
triangular method
v1 + v2 = v3
parallelogram method
v2 + v1 = v3
Note that all three ways produce the same resultant vector.
Instructions: Line them up “tip to tail.”
In other words, draw the first vector. Then at its tip, place the tail of the second vector.
The tip of this second vector is the tip of the resultant vector.
Vectors 2.1
Rev. F05
Mathematical Investigations IV
Name:
We can add as many vectors as we would like, e.g. v f  v1  v2  v 3  v 4
v2
v1 = < 1, 3 >
v3
v2 = < 4, 1 >
v3 = < 1, -1 >
v1
v4
v 4 = < -4, -2 >
vf
To add vectors analytically, we add their respective components.
Given two vectors v1  a1,b1  and v2  a2 ,b2  ,
v3  v1  v2  a1  a2, b1  b2 
For example, if v1 = <1, 3> and v2 = <4, -2>,
Then v1 + v2 = <5, 1>.
Verify this graphically:
Do a few yourself:
1. <-5, 1> + <2, 6> = <
,
>
 2  1   
2.      
2 1  
3. (cos2 iˆ + sin2 ĵ ) + (sin2 iˆ + cos2 ĵ ) = (
iˆ +
ĵ )
In arithmetic, subtraction is equivalent to the addition of the opposite of a given quantity.
The same is true of vectors.
Given v  a ,b  , the opposite of v is v  a,b 
or graphically
v
–v
Vectors 2.2
Rev. F05
Mathematical Investigations IV
Name:
To find v – w , we take v + (– w ). For example, for v = <1, 2>, w = <–3, 4>,
v + w = <–2, 6>
w
v – w = <4, –2>
v
–w
v+w
v–w
v
What is w - v ? Find the resultant vector both analytically and graphically.
Given the vectors v1   1, 3  , v2   4,1  , v3   1,  1  , v 4   4,  2  ,
find each of the following:
1.
v1  v2
2.
v1  v3  v4
3.
| v2  v 4 |
[Be careful!]
4.
(v4  v3 )

Vectors 2.3
Rev. F05
Mathematical Investigations IV
Name:
Let’s return to planes and boats. If a vehicle is moving in the presence of wind or current, the
resultant speed and direction in which the vehicle actually moves is the vector sum of the
velocity (speed and direction) in which the vehicle is heading and the velocity of the wind or
current.
For each of the following, draw a sketch, and then answer each question.
1.
If the wind blows at 20 mph due West and an airplane heads South at 400 mph, what is the
resultant speed and direction of the plane?
2.
A pilot wants to fly a course due NW at 350 mph. The wind is blowing at 25 mph due
West. In which direction and at what speed should the plane head, in order to fly the
desired course?
3.
A boat heads in the direction S30E at a speed of 25 knots. The tide runs N80E at 5 knots.
What is the resultant speed and direction of the ship?
Vectors 2.4
Rev. F05