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Vector Basics

April 14th, 2009 |

Author: Merrin

Up until now we have work with quantities that were scalar in nature. Scalars are quantities are one dimensional such as temperature or the area under a curve. Such quantities don’t point in a particular direction even though they can be positive or negative depending on your temperature system or convention for integration. In this chapter, we will study new objects that have both a numerical quantity and point in a particular direction for example in three dimensional space. These objects are called vectors.

Think of a one sided arrow pointing between two coordinates in three dimensional space as a vector. The origin of a vector can move, but the direction as well as the length must remain the same. The length of a vector is also referred to as the magnitude. A vector can be described by three numbers called components. Components are a measure of how much the vector points in each $x,y,z$ direction.

$\mathbf{V}=V_{x} \mathbf{i}+V_{y} \mathbf{j}+V_{z} \mathbf{k}$

The magnitude is the length the vector spans. We can find this quantity by the distance formula in three dimensions placing the origin of the vector at (0, 0, 0).

$\left|\mathbf{V}\right|=\sqrt{V_{x}^{2} +V_{y}^{2} +V_{z}^{2} }$

Since vectors are a new system, mentioning a few of the fundamental properties is helpful.

Property 1. The fundamental set of unit vectors each point in the direction of one of the cartesian axes.

$\mathbf{i} \quad \mathbf{j} \quad \mathbf{k}$

The unit vectors can be thought of as the identity vectors in each direction respectively since they have a length or magnitude of one. As a notation unit vectors will be written in lower case bold and other vectors will be written in capital case bold. A unit vector may correspond to its regular vector by

$\mathbf{r} = \cfrac{\mathbf{R}}{\left|\mathbf{R}\right|}$

Property 2. A vector can be defined as a sum of components multiplied by unit vectors

$\mathbf{V}=V_{x} \mathbf{i}+V_{y} \mathbf{j}+V_{z} \mathbf{k}$

Property 3. Two vectors can be added or subtracted according to an addition law.

$\mathbf{V}+\mathbf{W}=(V_{x} +W_{x})\mathbf{i} +(V_{y} +W_{y})\mathbf{j} + (V_{z} +W_{z})\mathbf{k}$

Property 4. Vector addition and subtraction is linear.

$c_{1} \mathbf{V}+c_{2} \mathbf{W}=(c_{1} V_{x} +c_{2} W_{x})\mathbf{i} +(c_{1} V_{y} +c_{2} W_{y})\mathbf{j} +(c_{1} V_{z} +c_{2} W_{z} )\mathbf{k}$

Property 5. Vector addition is commutative.

$\mathbf{A}+\mathbf{B}=\mathbf{C}=\mathbf{B}+\mathbf{A}$

Property 6. Vector addition is associative

$\mathbf{A}+(\mathbf{B}+\mathbf{C})=(\mathbf{A}+\mathbf{B})+\mathbf{C}$

Property 7. Multiplication by a number is commutative.

$c\mathbf{A}=\mathbf{A}c=\mathbf{B}$

Vectors can be multiplied by numbers which gives a stretched or shrunken vector. Vectors which are the same up to a multiplicative constant are collinear. If the constant is negative then the vectors are said to be antiparallel and if it is positive the vectors are said to be parallel.

Property 8. Multiplication by constants is associative

$b(c\mathbf{A})=(bc)\mathbf{A}$

Definition 1. A scalar field is a number at every point in a three dimensional space. Examples of scalar fields are temperature and density.

Definition 2. A vector field is a fixed vector at every point in a three dimensional space. Examples of vector fields are electric fields, magnetic fields, gravitational fields, and velocities of fluids. Each component of the vector field is a function of x, y, and z. We can write a vector field as

$\mathbf{V}(x,y,z)=V_{x} (x,y,z)\mathbf{i}+V_{y} (x,y,z)\mathbf{j}+V_{z} (x,y,z)\mathbf{k}$