Rules of vector algebra

Here we start by defining some rules of vector arithmetic using two vectors, $\boldsymbol{a}$ and $\boldsymbol{b}$. Note that we define $-\boldsymbol{a}$ to be the vector of the same magnitude as $\boldsymbol{a}$ but in the opposite direction.

Addition and subtraction

Given the vectors $\boldsymbol{a}=\left(a_{1}, a_{2}, a_{3}\right)$ and $\boldsymbol{b}=\left(b_{1}, b_{2}, b_{3}\right)$,

The diagram below shows the geometric interpretation of the addition of two vectors; it is known as the parallelogram law and it shows that vector addition is commutative, i.e.

The geometric interpretation of the subtraction of two vectors, $\boldsymbol{b}-\boldsymbol{a}$ is given below. The subtraction can be thought of as the addition of the vectors $\boldsymbol{b}$ and $-\boldsymbol{a}$ since $-\boldsymbol{a}$ has the same magnitude as $\boldsymbol{a}$ but in the opposite direction.

Multiplication by a scalar

For a vector $\boldsymbol{a}$ and a nonzero scalar $\lambda$, the vector $\lambda \boldsymbol{a}$ is the vector of magnitude $|\lambda| \times|\boldsymbol{a}|$ and in the direction of $\boldsymbol{a}$ if $\lambda>0$ or the opposite direction if $\lambda<0$. For $\boldsymbol{a}=\left(a_{1}, a_{2}, a_{3}\right)$, the scalar multiplication gives,

Using the two properties above, we can show what was stated in Definition 9.4: any 3D vector is a linear combination of the standard basis vectors in 3D. Starting from the vector $\boldsymbol{a}=\left(a_{1}, a_{2}, a_{3}\right)$, we can use the addition rule to rewrite as

and the scalar multiplication rule to express as,

Figure 9.2: Three-dimensional Cartesian coordinate system with axes $x, y, z$.

This shows that any vector can be written in terms of the standard basis vectors. Consider the Cartesian coordinates $x, y, z$ shown in Fig. 9.2. The unit vectors $\mathbf{i}, \mathbf{j}, \mathbf{k}$ are the standard basis vectors in Eq. (9.1). The vectors $\mathbf{i}, \mathbf{j}, \mathbf{k}$ are parallel to the $x, y, z$ axes, respectively.

The vector $\boldsymbol{r}=\left(a_{1}, a_{2}, a_{3}\right)$ that starts at the point $A=(0,0,0)$ and ends at $B=\left(a_{1}, a_{2}, a_{3}\right)$ is called a position vector. The vector $\boldsymbol{v}=(0,0,0)$ which has all its components equal to zero is called a zero vector, sometimes denoted by $\mathbf{0}, \overrightarrow{0}, \underline{0}$, etc.

The position vector with general coordinates $(x, y, z)$ is written as

where $x, y, z$ are the components of $\boldsymbol{r}$. The magnitude of $\boldsymbol{r}$ is given by

Example 9.1 Prove that the line joining a vertex of a parallelogram to the midpoint of the opposite side divides the diagonal in the ratio 1:2.

Solution Figure 9.3 shows a diagram of the parallelogram. We have

for some scalar $\mu$ and we want to show that $\mu=1 / 3$.

We define

for some scalar $\lambda$. Next, we consider

and we choose to express Eq. (9.2) in terms of a basis; we choose the position vectors $\overrightarrow{O A}$ and $\overrightarrow{O C}$ as the basis. By definition, we have

it follows that in Eq. (9.2), we are left to express $\overrightarrow{D P}$ and $\overrightarrow{O P}$ in terms of $\overrightarrow{O A}$ and $\overrightarrow{O C}$. We start with $\overrightarrow{O P}$ :

For $\overrightarrow{D P}$ :

Substituting Eqs. (9.3)-(9.5) in (9.2) and rearranging, yields

Comparing coefficients of $\overrightarrow{O A}$ and $\overrightarrow{O C}$ on both sides, we have $\mu=\lambda$ and $\mu=1 / 3$.

Figure 9.3: Diagram for question in Example 9.1.

Vector spaces

A vector is an element of a vector space. In broad terms, a vector space is a set of vectors together with rules for vector addition and scalar multiplication. Such operations must produce vectors in the space and satisfy certain conditions. Definition 9.5 gives said conditions.

A vector space $V$ is a set closed under finite vector addition and scalar multiplication. The scalars may come from any field $\mathbb{F}$, e.g. on real $\mathbb{F}=\mathbb{R}$ or complex $\mathbb{F}=\mathbb{C}$ vector spaces. For all $\boldsymbol{u}, \boldsymbol{v}$, and $\boldsymbol{w} \in V$ and $\lambda, \mu \in \mathbb{F}$, we have the following requirements:

(i) Commutativity

(ii) Associativity of vector addition

Additive identity

There exists the zero vector $\mathbf{0}$ such that for all $\boldsymbol{u}$,

(iv) Existence of additive inverse

For any $\boldsymbol{u}$ there exists $-\boldsymbol{u}$, such that,

(v) Associativity of scalar multiplication

(vi) Distributivity of scalar sums

(vii) Distributivity of vector sums

(viii) Scalar multiplication identity

A fun exercise is to use the rules given in Definition 9.5 to prove that for all $\boldsymbol{v} \in V$,