Many quantities in the real world do not simply exist as numbers.
When your car is moving, it has both speed and direction. To describe
the motion of the car, you need some way of writing this information
which gives both pieces of information. If you are giving directions
in a city, it is not enough to say, "walk four blocks", you have to say
walk 2 blocks north and 2 blocks west. Both magnitude and direction
are important if you want to get your point across.
A vector is a quantity
with both a magnitude and a direction, and it can be written either
as a magnitude and a direction (such as 60 miles per hour northwest)
or as coordinates (such as 2 blocks west and 2 blocks north).
Adding vectors is a fairly
common task. For example, suppose you are flying a plane. Your instruments
give you the speed of the plane with respect to the air around you, but
the wind is moving with respect to the ground as well! If you want to
know how fast you are moving with respect to the ground, you have
to determine your relative motion to the ground by adding your velocity
with respect to the wind to the wind's velocity with respect to the ground.
If the wind is directly at your back, this is easy, but suppose you have a
crosswind, how would you do this?
One way of doing this is to draw a picture. You can either
draw both vectors starting at the origin, and use them as the sides
of a parallelogram, or draw the first vector from the origin, and the
second vector starting at the tip of the first. These two graphical
techniques are known as the parallelogram and tip-to-tail methods,
respectively. The online model that goes along with this lesson
will let you use either method.
Suppose you are crossing a river at 8 meters per second, and the river
is flowing downstream at 6 meters per second. Assuming you are always swimming
directly towards the opposite shore, what would the resulting
speed and direction with respect to the river bank be?
Suppose you are in a plane flying directly north at a speed of 80
knots, and the wind is at your back, but blowing northwest at 20 knots.
What is the actual speed and direction your plane flies with respect to the
The drawback to the graphical methods is that they are only
as accurate as the person drawing and reading the graph. Also, consider
what happens when you have to add not just two vectors, but three, or four,
or twenty-thousand. What if you have an equation, and instead of
being able to draw the length of a vector, you need to leave it
as a variable?
Describe a situation where the use of a graphical method would
not be the best way to solve a vector addition problem.
When vectors are written in terms of x and y (and z for three dimensions)
coordinates, to add two vectors, you simply add up the coordinates.
This is easy enough to do for something like city blocks, but what
do you do if you do not have clear cut x and y coordinates? You have
to find the x and y coordinates for each vector, add components, and
if needed solve for the magnitude and direction of the resultant
Suppose you have a boat capable of traveling at 20 miles per hour. The
river is flowing at 5 miles per hour downstream. If you need to reach a dock
directly across from where you are starting, and the magnitude of
your boat's velocity with respect to the water is 20 miles per hour, how
would you set up an equation to solve for the direction to steer the boat in
in the fewest possible steps?
What direction would you steer the boat in?
How many steps did your solution take?
Compare with your classmates, who had the most efficient way of
getting the answer, and how did she/he do it?