In the chapter on vectors, we described the practical aspects of vectors. Also, people who have studied mechanics should be familiar with the force calculations which involved vectors.

In this section, we will describe vectors more abstractly—as mathematical objects. The first thing to do after one defines a new mathematical object is to specify its properties and the operations that we can perform on them. What can you do with numbers? I know how to add, subtract, multiply and divide numbers. The question, now, is to figure out the equivalent operations applied to vectors.

Consider two vectors $\vec{u}=(u_1,u_2,u_3) $ and $\vec{v}=(v_1,v_2,v_3)$, and assume that $\alpha$ is some number. We have the following properties:

\[ \begin{align} \alpha \vec{u} &= (\alpha u_1,\alpha u_2,\alpha u_3) \nl \vec{u} + \vec{v} &= (u_1+v_1,u_2+v_2,u_3+v_3) \nl \vec{u} - \vec{v} &= (u_1-v_1,u_2-v_2,u_3-v_3) \nl ||\vec{u}|| &= \sqrt{u_1^2+u_2^2+u_3^2} \nl \vec{u} \cdot \vec{v} &= u_1v_1+u_2v_2+u_3v_3 \nl \vec{u} \times \vec{v} &= (u_2v_3-u_3v_2,\ u_3v_1-u_1v_3,\ u_1v_2-u_2v_1) \end{align} \]

In the sections that follow we will see what these operations can do for us and what they imply.

The set of real numbers is denoted $\mathbb{R}$, and vectors consists of $d$ numbers, slapped together in a bracket. The numbers in the bracket are called components. If $d=3$, we will denote the set of vectors as: \[ ( \mathbb{R}, \mathbb{R}, \mathbb{R} ) \equiv \mathbb{R}^3 = \mathbb{V}(3), \] and similarly for more dimensions.

The notation $\mathbb{V}(n)$ for the set of $n$-dimensional vectors is particular to this section. It will be useful here as an encapsulation method, when we want to describe function signatures: what parameters it takes as inputs, and what outputs it produces. This section lists all the operations that take one or more elements of $\mathbb{V}(n)$ as inputs.

Addition and subtraction take two vectors as inputs and produce another vector as output. \[ +: \mathbb{V} \times \mathbb{V} \to \mathbb{V} \]

The addition and subtraction operations are performed component wise: \[ \vec{w}=\vec{u}+\vec{v} \qquad \Leftrightarrow \qquad w_{i} = u_i + v_i, \quad \forall i \in [1,\ldots,d]. \]

The scaling of a vector by a constant is an operation that has the signature: \[ \textrm{scalar-mult}: \mathbb{R} \times \mathbb{V} \ \to \ \mathbb{V}. \] There is no symbol to denote scalar multiplication—we just write the scaling factor in front of the vector and it is implicit that we are multiplying the two.

The scaling factor $\alpha$ multiplying the vector $\vec{u}$ is equivalent to this scaling factor multiplying each component of the vector: \[ \vec{w}=\alpha\vec{u} \qquad \Leftrightarrow \qquad w_{i} = \alpha u_i, \quad \forall i \in [1,\ldots,d]. \] For example, choosing $\alpha=2$ we obtain the vector $\vec{w}=2\vec{u}$ which is two times longer than the vector $\vec{v}$: \[ \vec{w}=(w_1,w_2,w_3) = (2u_1,2u_2,2u_3) = 2(u_1,u_2,u_3) = 2\vec{u}. \]

TODO copy over images from vectors chapter, and import other good passages

There are two ways to multiply vectors. The dot product: \[ \cdot: \mathbb{V} \times \mathbb{V}\ \to \mathbb{R}, \] \[ c=\vec{u}\cdot\vec{v} \qquad \Leftrightarrow \qquad c = \sum_{i=1}^d u_iv_i, \] and the cross product: \[ \times: \mathbb{V}(3) \times \mathbb{V}(3) \ \to \mathbb{V}(3) \] \[ \vec{w} = \vec{u} \times \vec{v} \qquad \Leftrightarrow \qquad \begin{array}{rcl} w_1 &=& u_2v_3-u_3v_2, \nl w_2 &=& u_3v_1-u_1v_3, \nl w_3 &=& u_1v_2-u_2v_1. \end{array} \] The dot product is defined for any dimension $d$. So long as the two inputs are of the same length, we can “zip” down their length computing the sum of the products of the corresponding entries.

The dot product is the key tool for dealing with projections, decompositions, and calculating orthogonality. It is also known as the scalar product or the inner product. Intuitively, applying the dot product to two vectors produces a scalar number which carries information about how similar the two vectors are. Orthogonal vectors are not similar at all, since no part of one vector goes in the same direction as the other, so their dot product will be zero. For example: $\hat{\imath} \cdot \hat{\jmath} = 0$. Another notation for the inner product is $\langle u | v \rangle \equiv \vec{u} \cdot \vec{v}$.

The cross product or vector product as it is sometimes called, is an operator which returns a vector that is perpendicular to both of the input vectors. For example: $\hat{\imath} \times \hat{\jmath} = \hat{k}$. Note the cross product is only defined for $3$-dimensional vectors.

The length of the vector $\vec{u} \in \mathbb{R}^d$ is computed as follows: \[ \|\vec{u}\| = \sqrt{u_1^2+u_2^2+ \cdots + u_d^2 } = \sqrt{ \vec{u} \cdot \vec{u} }. \] The length is number (always greater than zero) which describes the extent of the vector in space. The notion of length is a generalization of Pythagoras' formula for the length hypotenuse of a triangle given the lengths of the two sides (the components).

There exits more mathematically precise ways of talking about the intuitive notion of length. We could specify that we mean Euclidian length of the vector, or the ell-two norm $|\vec{u}|_2 \equiv ||u||$.

The first of these refers to the notion of a Euclidian space, which is the usual flat space that we are used to. Non-Euclidian geometries are possible. For example, the surface of the earth is spherical in shape and so when talking about lengths on the surface of the earth we will need to use spherical length, not Euclidian length. The name ell-two norm refers to the fact that we raise each coefficient to the second degree and then take the square root when computing the length. An example of another norm is the ell-four norm which is defined as the fourth root of the sum of the components raised to the fourth power: $|\vec{u}|_4 \equiv \sqrt[4]{u_1^4+u_2^4+u_3^4}$.

Often times in physics, we denote the length of a vector $\vec{r}$ simply as $r$. Another name for length is magnitude.

Note how the length of a vector can be computed by taking the dot product of the vector with itself and then taking the square root: \[ \|\vec{v}\| = \sqrt{ \vec{v} \cdot \vec{v} }. \]

Given a vector $\vec{v}$ of any length, we can build a unit vector in the same direction by dividing $\vec{v}$ by its length: \[ \hat{v} = \frac{\vec{v}}{ ||\vec{v}|| }. \]

Unit vectors are useful in many contexts. In general, when we want to specify a direction in space, we use a unit vector in that direction.

If I give you a direction $\hat{d}$ and some vector $\vec{v}$ and ask you how much of $\vec{v}$ is in the $\hat{d}$-direction, then the answer is computed using the dot product: \[ v_d = \hat{d} \cdot \vec{v} \equiv \| \hat{d} \| \|\vec{v} \| \cos\theta = 1\|\vec{v} \| \cos\theta, \] where $\theta$ is the angle between $\vec{v}$ and $\hat{d}$. We used this formula a lot in physics when we were computing the $x$-component of a force $F_x = \|\vec{F}\|\cos\theta$.

We define the projection of a vector $\vec{v}$ in the $\hat{d}$ direction as follows: \[ \Pi_{\hat{d}}(\vec{v}) = v_d \hat{d} = (\hat{d} \cdot \vec{v})\hat{d}. \]

If the direction is specified by a unit vector $\vec{d}$ which is not unit length, then the formula becomes: \[ \Pi_{\vec{d}}(\vec{v}) = \left(\frac{ \vec{d} \cdot \vec{v} }{ \|\vec{d}\|^2 } \right) \vec{d}. \] The division by the length squared is necessary in order to turn the vectors $\vec{d}$ into unit vectors $\hat{d}$ as required but the projection formula: \[ \Pi_{\vec{d}}(\vec{v}) = (\vec{v}\cdot\hat{d}) \:\hat{d} = \left(\vec{v}\cdot \frac{\vec{d}}{\|\vec{d}\|}\right) \frac{\vec{d}}{\|\vec{d}\|} = \left(\frac{\vec{v}\cdot\vec{d}}{\|\vec{d}\|^2}\right)\vec{d}. \]

This section was a review of the properties of $d$-dimensional vectors. These are simply ordered tuples (lists) of $d$ coefficients. It is important to think of vectors as mathematical objects and not as coefficients. Sure, all the vector operations boil down to manipulations of the coefficients in the end, but vectors are most useful (and best understood) if you think of them as one thing that has components rather than focussing on the components.

In the next section we will learn about another mathematical object: the matrix, which is nothing more than a two-dimensional array (a table) of numbers. Again, you will see, that matrices are more useful when you think of their properties as mathematical objects rather than focussing on the individual numbers that make up their rows and columns.