Introduction to Vectors

# Introduction to Vectors

We will now look into an important type of object known as a vector. If you are already familiar with vectors, then the material coming up will not be new, and in fact, all of it can be found on the Linear Algebra page on this site. Nevertheless though, we will review this material as we will need vectors shortly.

 Definition: A Vector denoted $\vec{u}$ or $\mathbf{u}$ is an object that has both a magnitude and direction. The Magnitude denoted $\| \vec{u} \|$ of a vector is its length or size, while the Direction of the vector is the position that the vector points towards.

Algebraically, a vector in $\mathbb{R}^n$ can be represented in terms of its components as $\vec{u} = (u_1, u_2, ..., u_n)$. For example, the vector $(1, 2) \in \mathbb{R}^2$ represents a vector that has moved $1$ along the $x$-axis and $2$ along the $y$-axis.

Vectors in $\mathbb{R}$, $\mathbb{R}^2$ and $\mathbb{R}^3$ can also be represented geometrically with straight arrows, though representing vectors as arrows in higher dimensions is difficult.

The starting point of the vector is known as the Initial Point, while the end point of the vector is known as the Terminal Point. Therefore, a vector $\vec{u}$ that is defined to start at point $P$ and end at point $Q$ can be represented as $\vec{u} = \vec{PQ}$. We should note that a vector such as $(1, 2, 3)$ can be placed anywhere provided that this vector goes right $1$, up $2$ and forward $3$, however, for convenience, we often times prefer to place the initial point of a vector at the origin. These type of vectors have a special name.

 Definition: $\vec{r} \in \mathbb{R}^n$ is said to be a Position Vector if the initial point of $\vec{r}$ is the origin.

## Operations on Vectors

We will now look at two important operations defined for vectors, namely vector addition and multiplication by a scalar.

 Definition: Given two vectors $\vec{u}, \vec{v} \in \mathbb{R}^n$, Vector Addition is defined as $\vec{u} + \vec{v} = (u_1, u_2, ..., u_n) + (v_1, v_2, ..., v_n) = (u_1 + v_1, u_2 + v_2, ..., u_n + v_n)$.

For example, consider the vectors $\vec{u} = (1, 2, 3)$ and $\vec{v} = (7, 4, 6)$ in $\mathbb{R}^3$. Then $\vec{u} + \vec{v} = (1, 2, 3) + (7, 4, 6) = (1 + 7, 2 + 4, 3 + 6) = (8, 6, 9)$. The following diagram geometrically describes vector addition in $\mathbb{R}^2$. Definition: Given a vector $\vec{u} \in \mathbb{R}^n$, Multiplication by a Scalar $k$ is defined as $k\vec{u} = k(u_1, u_2, ..., u_n) = (ku_1, ku_2, ..., ku_n)$.

For example, if $k = 2$ and we have the vector $\vec{u} = (5, 3, 2, 1)$ in $\mathbb{R}^4$ then $k\vec{u} = 2(5,3,2,1) = (10,6,4,2)$. Geometrically, if we multiply a vector by a positive scalar $k > 0$ then the magnitude of $\vec{u}$ is multiplied by $k$. If we multiply a vector by a negative scalar $k < 0$ then the magnitude of $\vec{u}$ is multiplied by $\mid k \mid$ and $k\vec{u}$ is pointed in the opposite direction to that of $\vec{u}$. The diagram below represents vector multiplication by a scalar $k > 0$ in $\mathbb{R}^2$. With the operations of vector addition and scalar multiplication, there is a huge list of properties we can easily derive as outlined in the following theorem.

 Theorem 1: Let $\vec{u}, \vec{v}, \vec{w} \in \mathbb{R}^n$ and let $k$ and $l$ be scalars. Then the following properties hold: a) $\vec{u} + \vec{v} = \vec{v} + \vec{u}$ (Commutativity of Scalar Addition). b) $\vec{u} + (\vec{v} + \vec{w}) = (\vec{u} + \vec{v}) + \vec{w}$ (Associativity of Scalar Addition). c) $\vec{u} + \vec{0} = \vec{0} + \vec{u} = \vec{u}$ (Addition by the Zero Vector $\vec{0} = (0, 0, ..., 0) \in \mathbb{R}^n$). d) $\vec{u} + (-\vec{u}) = (-\vec{u}) + \vec{u} = \vec{0}$. e) $1 \vec{u} = \vec{u}$. f) $k(l\vec{u}) = (kl) \vec{u}$. g) $(k + l)\vec{u} = k \vec{u} + l \vec{u}$. h) $k(\vec{u} + \vec{v}) = k\vec{u} + k \vec{v}$.

We will now prove any part to Theorem 1 as the proofs are rather trivial, but the reader is encouraged to attempt a few of them if they are not familiar with vectors or check out the Linear Algebra page for their proofs.

## Magnitude of a Vector

In the definition of a vector we described a vector $\vec{u}$ having a magnitude denoted $\| \vec{u} \|$. Many texts will actually denote the magnitude with single vertical bars, but we will try not to use this notation as to not confuse it with absolute value bars.

Now, computing the magnitude of a vector is relatively easy with the following formulas.

 Magnitude of a Vector in 2-Space: If $P(x_1, y_1)$ and $Q(x_2, y_2)$ define a vector $\vec{PQ}$ in $2$-space, then the magnitude of $\vec{PQ}$ can be calculated with the formula $\mid \vec{PQ} \mid = \sqrt{ (x_2 - x_1)^2 + (y_2 - y_1)^2}$.
 Magnitude of a Vector in 3-Space: If $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$ define a vector $\vec{PQ}$ in $3$-space then the magnitude of $\vec{PQ}$ can be calculated with the formula $\mid \vec{PQ} \mid = \sqrt{ (x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$. A vector of length $1$ is said to be a Unit Vector and $\vec{u}_{\mathrm{unit}} = \frac{1}{\| v \|} \vec{v}$.

The formula for the magnitude of a vector in $\mathbb{R}^n$ can be derived in a similar manner.

For example, consider the point $P(1, 2)$ and $Q(2, 3)$. Then the magnitude of the vector $\| \vec{PQ} \| = \sqrt{(2 - 1)^2 + (3 - 2)^2} = \sqrt{1^2 + 1^2} = \sqrt{2}$. Another example is computing the magnitude of $\vec{u} = (1, 4)$. We can choose the origin as our initial point, and so $\| \vec{u} \| = \sqrt{1^2 + 4^2} = \sqrt{17}$.

## Standard Basis Vectors

In $\mathbb{R}^2$ we have two coordinates axes, and in $\mathbb{R}^3$ we have three coordinate axes. Any vector $\vec{u}$, say in $\mathbb{R}^2$ can be represented as an ordered $2$-tuple which describes how many units this vector goes right or left, and how many units this vector goes up or down. In actuality, what we are doing is taking a linear combination of the vectors $(1, 0)$ and $(0, 1)$. For example, if $\vec{u} = (2, 5)$ then $\vec{u} = 2(1, 0) + 5(0, 1)$. In fact, any vector $\vec{u} = (a, b) = a(1, 0) + b(0, 1)$. We thus call the vectors $(1, 0)$ and $(0, 1)$ the standard basis vectors of $\mathbb{R}^2$.

 Definition: In $\mathbb{R}^2$ the standard basis vectors are $\vec{i} = (1, 0)$ and $\vec{j} = (0, 1)$. In $\mathbb{R}^3$, the standard basis vectors are $\vec{i} = (1, 0, 0)$, $\vec{j} = (0, 1, 0)$ and $\vec{k} = (0, 0, 1)$. In $\mathbb{R}^n$, the standard basis vectors are $\vec{e_1} = (1, 0, 0, ..., 0)$, $\vec{e_2} = (0, 1, 0, ..., 0)$, …, $\vec{e_n} = (0, 0, ..., 0, 1)$.

Often times for brevity we will condense the vector notation down for some vector, say $(x, y, z) \in \mathbb{R}^3$ as $x\vec{i} + y\vec{j} + z\vec{k}$. For example, $(1, 2, 3) = \vec{i} + 2\vec{j} + 3\vec{k}$.