Vectors and the Parallelogram Method

Lesson 4

Kendalyn Paulin

Everyday Examples of Vectors

If you were an air traffic controller, how would you describe the movement of an airplane?

What about describing the movement of a car?

NTCM example

Vector Activity This activity is an example of a car being

represented by a vector. Let’s explore some interesting concepts

using this site!

In order to show direction and speed of an object, vectors are used.

A vector is a mathematical quantity that has both a magnitude (length) and direction.

It is often denoted as a lowercase bold face letter, such as v.

Sometimes an arrow will be drawn over the letter. For our lessons, we will use a lowercase bold

letter or italicized uppercase letters.

A vector has an initial point, and a terminal point. It resembles a ray.

This is vector PQ

Initial Point

P

Terminal Point

Q

It does not matter where a vector is located in a plane, as long as it maintains the same direction and magnitude.

For example, all the vectors below are equal.

Equivalent Vectors

To prove two vectors are equivalent, they first both must be pointing in the same direction and have the same slope.

How could we physically prove that two vectors are equivalent?

ONE WAY would be to plot the vector on a coordinate grid and use the equation :

12

12

xx

yyslope

Equivalent Vectors

Second, they both must have the same magnitude (length).

How could we show they have the same length?

ONE WAY would be to use the Distance Formula: 2

122

12 )( yyxxlegnth

Equivalent Vectors

You could also overlay the two vectors. If they are the same, then they are equivalent.

You could also transpose the coordinates of one vector and see if they match the other vector.

Any other ideas?

Scalar Multiplication: A scalar (k) will make a vector k times as long. If k is positive, k*v has the same direction as v. If k is negative, k*v has the opposite direction of v.

v (1/2)v 2v -v

Vector Addition When we add two vectors together, we can move the

vectors wherever we want in the plane as long as the direction and length stay the same.

We want to place the tail of one vector on top of the head of another vector.

It does not matter which vector you decide to use first.

=+

Once these two vectors are added, a resultant vector can be drawn connecting the tail of the first vector and the head of the second vector, creating a triangle. The black vector in this example is the resultant vector of red vector + blue vector.

Parallelogram Method When adding two vectors that share the same

tail… Draw the first vector again by placing its tail on

the head of the second vector. Then draw the second vector by placing its tail on the head of the first vector. The diagonal is the resultant vector.

You are not changing the direction or the magnitude.

THIS IS BETTER EXPLAINED USING PICTURESNEXT SLIDE

Parallelogram Method

+

The Black Vector represents the RESULTANT VECTOR of the red and blue vectors.

p

q

r

OM

Representing the sum geometrically…

Vectors OM+MN=ON

which is equivalent to p+q=rNK

Scale Drawing and Direct Measurement

Another way to find the magnitude and direction of a resultant vector is by using scale drawing and direct measurement.

Direction of a Vector Say we want to tell someone

what direction an object is moving.

We can use angle measurements to determine this value.

Take this resultant vector as an example…

The angle theta is the angle between the positive x axis and the resultant vector.

This is the direction of the resultant vector

Direction of a Resultant Vector

You can also use your knowledge of tangent to find a resultant vector.

Move your resultant vector so that the tail is at (0,0). Then use tan θ= opposite/adjacent and solve for θ.

Example on next slide.

θ

tan θ= opposite/adjacenttan θ= 5/3θ=tan-1(5/3)Θ=59 degrees

Let’s use a worksheet to help us understand how to use scale drawing and direct

measurement