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Introduction to Vector Mathematics


Introduction to Vector Mathematics

Components of a Vector

Andrew Zimmerman Jones
This is a basic, though hopefully fairly comprehensive, introduction to working with vectors. Vectors manifest in a wide variety of ways, from displacement, velocity and acceleration to forces and fields. This article is devoted to the mathematics of vectors; their application in specific situations will be addressed elsewhere.

Vectors & Scalars

In everyday conversation, when we discuss a quantity we are generally discussing a scalar quantity, which has only a magnitude. If we say that we drive 10 miles, we are talking about the total distance we have traveled. Scalar variables will be denoted, in this article, as an italicized variable, such as a.

A vector quantity, or vector, provides information about not just the magnitude but also the direction of the quantity. When giving directions to a house, it isn't enough to say that it's 10 miles away, but the direction of those 10 miles must also be provided for the information to be useful. Variables that are vectors will be indicated with a boldface variable, although it is common to see vectors denoted with small arrows above the variable.

Just as we don't say the other house is -10 miles away, the magnitude of a vector is always a positive number, or rather the absolute value of the "length" of the vector (although the quantity may not be a length, it may be a velocity, acceleration, force, etc.) A negative in front a vector doesn't indicate a change in the magnitude, but rather in the direction of the vector.

In the examples above, distance is the scalar quantity (10 miles) but displacement is the vector quantity (10 miles to the northeast). Similarly, speed is a scalar quantity while velocity is a vector quantity.

A unit vector is a vector that has a magnitude of one. A vector representing a unit vector is usually also boldface, although it will have a carat (^) above it to indicate the unit nature of the variable. The unit vector x, when written with a carat, is generally read as "x-hat" because the carat looks kind of like a hat on the variable.

The zero vector, or null vector, is a vector with a magnitude of zero. It is written as 0 in this article.

Vector Components

Vectors are generally oriented on a coordinate system, the most popular of which is the two-dimensional Cartesian plane. The Cartesian plane has a horizontal axis which is labeled x and a vertical axis labeled y. Some advanced applications of vectors in physics require using a three-dimensional space, in which the axes are x, y, and z. This article will deal mostly with the two-dimensional system, though the concepts can be expanded with some care to three dimensions without too much trouble.

Vectors in multiple-dimension coordinate systems can be broken up into their component vectors. In the two-dimensional case, this results in a x-component and a y-component. The picture to the right is an example of a Force vector (F) broken into its components (Fx & Fy). When breaking a vector into its components, the vector is a sum of the components:

F = Fx + Fy
To determine the magnitude of the components, you apply rules about triangles that are learned in your math classes. Considering the angle theta (the name of the Greek symbol for the angle in the drawing) between the x-axis (or x-component) and the vector. If we look at the right triangle that includes that angle, we see that Fx is the adjacent side, Fy is the opposite side, and F is the hypotenuse. From the rules for right triangles, we know then that:
Fx / F = cos theta and Fy / F = sin theta

which gives us

Fx = F cos theta and Fy = F sin theta

Note that the numbers here are the magnitudes of the vectors. We know the direction of the components, but we're trying to find their magnitude, so we strip away the directional information and perform these scalar calculations to figure out the magnitude. Further application of trigonometry can be used to find other relationships (such as the tangent) relating between some of these quantities, but I think that's enough for now.
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