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Vectors

Ordered Pairs

Have you ever seen an X/Y Coordinate diagram before? Sure you have. The vertical axis is the "Y-Axis" and the horizontal axis is the "X-Axis". Both axes are perpendicular (intersect at ninety-degrees) to each other. In addition to seeing an X/Y Coordinate diagram you have also probably plotted points before as well. For example, you have probably plotted points such as A (+3, +6) before. Where the point "A" is located three units along the X-Axis and 6 Units along the Y-Axis. What if I told you that a vector had an Ordered Pair of A (+3, +6)? What this means is that the Vector A consists of components +3 in the positive X-direction and +12 in the positive Y-direction. By components we mean that Vector A is comprised of both an X-component and a Y-component. The meaning of a vector consisting of an Ordered Pair is that it contains components that lie in both the X and Y axes. See the example below of how to signify the Vector A given above:

Graph of Vector A

How do we find Vector A? We know that Ax and Ay intersect at a right angle. Thus we have a right triangle. Knowing this information is very powerful because we know a lot about right triangles. We know that the sum of the squares of the sides of a right triangle are equal to the square of the hypotenuse. More commonly known as the Pythagoream Theorem:

C2 = A2 + B2

To find the magnitude of A we can say:

A2 = Ax2 + Ay2

How do we find the direction? We also know that there is a relationship between the sides and hypotenuse of any right triangle. This relationship is also related to the angle between the each of the sides and hypotenuse. By using the trigonometric functions Sine, Cosine, and Tangent we can determine the angle between any two sides (and hypotenuse) if we know the magnitude (length) of the sides (and hypotenuse).

**Thus the magnitude of Vector A can be deduced using the Pythagorean theorem and the Trigonometric functions can be used to deduce the direction of Vector A**

Keep in mind that this we are just talking now about one particular vector. This method can be done for any vector with an Ordered Pair.

Do you think that there is a way to determine the Resultant vector if we are given two (or more than two) Ordered Pair Vectors? Of course, in fact it is very easy. Remember we said that a Resultant vector is just the sum of the vectors given. In other words R = A + B+ C... and we now know that Vector A (or B or C, etc.) consists of the vector sum of its component vectors in the X-Direction and Y-Direction or A2 = Ax2 + Ay2. Well now we can say that the Resultant Vector R, using Ordered Pairs, consists of the sum of its X and Y-Components. How do we get the sum of its X and Y-Components? By summing the X-Components of all of the vectors given (A, B, C, etc) and summing the Y-Components of all of the vectors a given. See the example given below for two vectors, A (+3, +6) and B (-5, -2).

Graph of Adding Vectors A and B

To find the Resultant Vector R it is very convenient and organized to set up the table given below:

Vector X-Coordinate Y-Coordinate
Vector A +3 +6
Vector B -5 -3
Resultant (R) -2 +3

Thus the Resultant Vector R has the Ordered Pair R (-2, +4). Take a look at the green vector given above in the diagram. Do the vector components correct in the diagram as compared to the Ordered Pair given? They are in fact the same. So how do we find the magnitude and direction? Pythagorean Theorem --> Magnitude, Trigonometric Function --> Direction.