Vectors are used for depicting direction and magnitude of motion, force, or other changes. Vectors are displayed on graphs and with numbers. Vectors are explained with the following three examples, to show you some of the diversity of vector applications, as well as give you ideas of how vectors are related to your kite.

A honeybee is travelling from rose to clover. As an entomologist, you are interested in documenting its pollination journey.

A kite is being pulled by the wind along its bridle line at a strong force (you can feel it tugging). You are interested in determining how much lift and drag your kite must have, based on this force.

The Coast Guard's gauge for water speed and direction have broken, so they need to determine the water speed and direction, using the known driving speed of the boat and their starting and stopping buoy positions.

A vector looks a lot like a line segment when displayed on a graph. The graphical display of a vector can help someone better understand the direction and magnitude of the vector, otherwise displayed with numbers below. Often arrows are added to summarize the direction of travel. The starting position can also be labeled (x1,y1), and later points (x2, y2) and so on.

The bee starts at the rose and ends at the clover. If the rose is at (0, 0) (staring point or origin), and the clover is at (5, -13) (5 feet to the right and 13 feet down from the rose), the path can be graphed as follows.

Notice that the journey could be broken down into vertical movement and horizontal movement. This graph is useful for summarizing a vector with numbers. Breaking the journey into the two movements creates a right triangle, which we can label as follows. Thinking about vectors in terms of right triangles is very useful for more advanced vector problems, like those in examples 2 and 3. Note that the vertical and horizontal components connect end-to-end or (arrowhead-to-arrowhead) to create the diagonal (hypotenuse) vector. "Adding" vectors in this way is very common and is called creating a dot product. In other words, Horiz o Vert = Diag. Notice that NO multiplication is involved in creating a dot product, despite its name. The symbol in the equation above is not a multiplication symbol, but an open dot.

Example 2: A spring scale is attached to the kite's bridle and measures 2 Newtons. You measure the bridle as being 45 degrees relative to the horizon. Lift is vertical to the horizon and pulling upwards, and drag is parallel to the horizon and pulling in the same direction as the wind (away from you). You can plot the vector for force along your string and associated lift and drag vectors, as follows.

To find the lift and drag vectors, you can recall that a 45-45-90 degree right triangle has side lengths of x and a hypotenuse of x times the square root of 2. A 45-45-90 degree triangle is also isosceles, so the side lengths squared should add to make the hypotenuse length squared, using Pythagorean Theorem: x² + x² = 2² Newtons². Trigonometry could also be used to solve this.
Now the graph can be labeled completely.

Example 3:The buoy the Coast Guard starts at is at (20,20) (20 meters up from and to the right of the dock). The buoy the boat ends at is at (250, 120) (250 meters to the right of and 120 meters up from the dock). The boat steers directly up (no right or left steering) and moves at 100 meters per minute for one minute. Two vectors can be plotted here - actual movement of the boat (connect the dots between the buoys) and the upwards movement of the boat that is intended. Remember that the Coast Guard wants to know how fast the water is moving and in what direction, so that will be the third vector in the right triangle formed.

To find the third vector, simple coordinate geometry is the easiest method to apply. When it is solved for, the completed graph looks like the one below. Notice that the units chosen are in meters per minute, as the speed of the water is what is desired. (It should be noted that water only moves in the horizontal vector, because the boat's motor took care of the entire vertical vector. A more complex problem would be if the boat's motor did not cover the entire vertical distance.)

There are two kinds of number displays.
(1) Direction.
Direction is often thought of as N, S, E, or W. Direction can become more specific by combining directions (e.g. NW, SE), or providing angle measurements. Vectors are a very precise way of providing direction, in terms of a horizontal and vertical component. How nice, that we have pictures of these components above! Direction is shown as a coordinate pair, (x, y), further described below.

direction = (change in horizontal movement, change in vertical movement)

change in movement = ending position - starting position

(2) Magnitude (Distance)
Magnitude or distance can be determined with the Pythagorean Theorem, trigonometry, special triangle rules, and the distance formula, as you saw in the examples above. The distance formula is:

D = square root ( (x₂ -x₁)² + (y₂-y₁)²)

D = square root (change in horizontal movement ² + change in vertical movement²)

Example 1: I used the graph to help me solve. Additionally, work is shown if the graph had not been produced.

	Direction	= (5-0, -13-0)
		= (5, -13)
	Magnitude	= square root (52 + -132)
		= square root (25 + 169)
		= square root (194)

Example 2: Three vectors are described. They will be labeled as V_x (horizontal vector), V_y (vertical vector), and V_xoy (diagonal vector or dot product).

Vx
	Direction	= ((square root(2)) - 0, 0)
		= ((square root(2)), 0)
	Magnitude	= square root ((square root(2))2 + 02)
		= square root (2) Newtons
Vy
	Direction	= (0, (square root (2)) - 0)
		= (0, (square root (2)))
	Magnitude	= square root (02 + (square root(2))2)
		= square root (2) Newtons
Vxoy
	Direction	= ((square root (2)) - 0,(square root (2)) - 0)
		= ((square root (2)), (square root (2)))
	Magnitude	= square root ((square root(2))2 + (square root(2))2)
		= 2 Newtons

Example 3: Three vectors are also described here, labeled as in example 2.

You Decide Intro
You Decide Scenario