\(\mathbb{R}^2\) as a set of points
We write \(\mathbb{R}^2\) to denote the set of all points in the plane.
A point is represented by an ordered pair of real numbers \((x, y)\).
The numbers \(x\) and \(y\) are called the coordinates of the point.
This is the usual way of thinking about points in the plane. You’ve known this your entire mathematical life.
However, sometimes it is convenient to think of \(\mathbb{R}^2\) in a different way:
\(\mathbb{R}^2\) as a set of vectors
We write \(\mathbb{R}^2\) to denote the set of all vectors in the plane.
Such a vector is represented as a \(2\)-dimensional column vectors written as \[ \begin{bmatrix} x \\ y \end{bmatrix} \] where \(x\) and \(y\) are real numbers.
The numbers \(x\) and \(y\) are called the components of the vector.
Visually, we can think of a vector \(\begin{bmatrix} x \\ y \end{bmatrix}\) as an arrow with its tail at the origin \((0,0)\) and its head at the point \((x,y)\).
Which way we think of \(\mathbb{R}^2\) (as points or as vectors) depends on the context. We will go back and forth between these two interpretations repeatedly.
Vector addition/subtraction
Given two vectors \[\mathbf{u} = \begin{bmatrix} u_1 \\ u_2 \end{bmatrix} \quad \text{and} \quad \mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \end{bmatrix},\] their sum/difference \(\mathbf{u} \pm \mathbf{v}\) is defined as \[\mathbf{u} \pm \mathbf{v} = \begin{bmatrix} u_1 \pm v_1 \\ u_2 \pm v_2 \end{bmatrix}.\]
Scalar multiplication
Given a vector \[\mathbf{u} = \begin{bmatrix} u_1 \\ u_2 \end{bmatrix}\] and a scalar (real number) \(c\), the scalar multiple \(c\mathbf{u}\) is defined as \[c\mathbf{u} = \begin{bmatrix} cu_1 \\ cu_2 \end{bmatrix}.\]
Properties of vector addition and scalar multiplication
Let \(\mathbf{u}\), \(\mathbf{v}\), and \(\mathbf{w}\) be vectors in \(\mathbb{R}^2\), and let \(a\) and \(b\) be scalars. Then the following properties hold:
Zero vector
Practice with vector addition, subtraction, and scalar multiplication
Given the vectors \[ \mathbf{u} = \begin{bmatrix} 2 \\ 3 \end{bmatrix} \quad \text{and} \quad \mathbf{v} = \begin{bmatrix} -1 \\ 4 \end{bmatrix}, \] compute the following:
Then, draw the vectors \(\mathbf{u}\) and \(\mathbf{v}\) in \(\mathbb{R}^2\), and the results of each operation.
Magnitude of a vector
The norm (or magnitude) of a vector \[ \mathbf{u} = \begin{bmatrix} u_1 \\ u_2 \end{bmatrix} \] is denoted by \(\lVert \mathbf{u}\rVert\) and is defined as \[ \lVert \mathbf{u}\rVert = \sqrt{u_1^2 + u_2^2}. \]
A vector is called a unit vector if its norm is \(1\).
a) Geometric interpretation of norms
Explain why the norm of a vector is its length, interpreting the vector as an arrow in the plane.
Given a vector \(\mathbf{u}\) and a scalar \(c\), explain why \(\lVert c\mathbf{u} \rVert = |c| \lVert \mathbf{u} \rVert\). Does this make sense geometrically?
b) Practice with computing norms
Given the vectors \[ \mathbf{a} = \begin{bmatrix} 3 \\ 4 \end{bmatrix} \quad \text{and} \quad \mathbf{b} = \begin{bmatrix} -1 \\ 2 \end{bmatrix}, \]
compute the following:
Algebraic definition of the dot product
Given two vectors \[\mathbf{u} = \begin{bmatrix} u_1 \\ u_2 \end{bmatrix} \quad \text{and} \quad \mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \end{bmatrix},\] their dot product \(\mathbf{u} \cdot \mathbf{v}\) is defined as \[\mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2.\]
The dot product is also called an inner product and is written alternatively as \(\langle \mathbf{u}, \mathbf{v} \rangle\).
We will switch between both notations \(\mathbf{u} \cdot \mathbf{v}\) and \(\langle \mathbf{u}, \mathbf{v} \rangle\) very often!
The above definition is called the algebraic definition of the dot product since it only involes algebraic operations on the components of the vectors.
There is also a geometric definition of the dot product, which relates it to the angle between the two vectors and their norms.
Geometric definition of the dot product
Given two vectors \(\mathbf{u}\) and \(\mathbf{v}\), the dot product can also be defined as \[\mathbf{u} \cdot \mathbf{v} = \lVert \mathbf{u} \rVert \lVert \mathbf{v} \rVert \cos \theta,\] where \(\theta\) (\(0 \leq \theta \leq \pi\)) is the angle between the two vectors.
Algebraic and geometric properties of the dot product
The dot product is commutative: \(\mathbf{u} \cdot \mathbf{v} = \mathbf{v} \cdot \mathbf{u}\).
The dot product is distributive over vector addition: \(\mathbf{u} \cdot (\mathbf{v} + \mathbf{w}) = \mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{w}\).
The dot product is bilinear: for any scalar \(c\), \((c\mathbf{u}) \cdot \mathbf{v} = c(\mathbf{u} \cdot \mathbf{v})\).
If \(\mathbf{u}\) and \(\mathbf{v}\) are nonzero, then they are orthogonal (i.e., perpendicular) if and only if \(\mathbf{u} \cdot \mathbf{v} = 0\).
a) Practice with computing dot/inner products
Given the vectors \[ \mathbf{p} = \begin{bmatrix} 1 \\ 2 \end{bmatrix} \quad \text{and} \quad \mathbf{q} = \begin{bmatrix} 3 \\ 4 \end{bmatrix}, \] compute the following:
b) Geometric interpretation of the dot/inner product
Explain why the algebraic and geometric definitions of the inner product \(\langle \mathbf{u} , \mathbf{v}\rangle\) are equivalent. For simplicity, you may assume that \(\mathbf{u}\) points along the x-axis, i.e., \[\mathbf{u} = \begin{bmatrix} u_1 \\ 0 \end{bmatrix}.\]
Standard basis vectors in \(\mathbb{R}^2\)
The standard basis vectors in \(\mathbb{R}^2\) are denoted by \(\mathbf{e}_1\) and \(\mathbf{e}_2\), and are defined as \[ \mathbf{e}_1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \quad \mathbf{e}_2 = \begin{bmatrix} 0 \\ 1 \end{bmatrix}. \]
Theorem: Resolving vectors into standard basis vectors
Any vector
\[ \mathbf{u} = \begin{bmatrix} u_1 \\ u_2 \end{bmatrix} \]
in \(\mathbb{R}^2\) can be expressed as a linear combination of the standard basis vectors (i.e., a sum of scalar multiples of the basis vectors) as:
\[\mathbf{u} = u_1 \mathbf{e}_1 + u_2 \mathbf{e}_2.\]
Practice with basis vectors
Given any vector \(\mathbf{u}\), explain why \[ \mathbf{u} = \langle \mathbf{u}, \mathbf{e}_1 \rangle \mathbf{e}_1 + \langle \mathbf{u}, \mathbf{e}_2 \rangle \mathbf{e}_2. \]
Take care to notice that there are three vector operations happening in this equation: inner products, scalar multiplication, and vector addition.
\(\mathbb{R}^3\) as a set of points
We write \(\mathbb{R}^3\) to denote the set of all points in \(3\)-dimensional space.
A point is represented by an ordered triple of real numbers \((x, y, z)\).
The numbers \(x\), \(y\), and \(z\) are called the coordinates of the point.
\(\mathbb{R}^3\) as a set of vectors
We write \(\mathbb{R}^3\) to denote the set of all vectors in \(3\)-dimensional space.
Such a vector is represented as a \(3\)-dimensional column vectors written as \[ \begin{bmatrix} x \\ y \\ z \end{bmatrix} \] where \(x\), \(y\), and \(z\) are real numbers.
The numbers \(x\), \(y\), and \(z\) are called the components of the vector.
All the definitions and properties of vector algebra, norms, and dot products that we discussed for \(\mathbb{R}^2\) carry over to \(\mathbb{R}^3\) with only minor modifications.
For exmaple, given two vectors \[\mathbf{u} = \begin{bmatrix} u_1 \\ u_2 \\ u_3 \end{bmatrix} \quad \text{and} \quad \mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \\ v_3 \end{bmatrix},\] their sum/difference is defined as \[\mathbf{u} \pm \mathbf{v} = \begin{bmatrix} u_1 \pm v_1 \\ u_2 \pm v_2 \\ u_3 \pm v_3 \end{bmatrix}.\]
Their norm is defined as \[\lVert \mathbf{u} \rVert = \sqrt{u_1^2 + u_2^2 + u_3^2}.\]
Their inner product is defined as \[\langle \mathbf{u}, \mathbf{v} \rangle = u_1v_1 + u_2v_2 + u_3v_3.\]
All the properties we discussed for vectors in \(\mathbb{R}^2\) also hold in \(\mathbb{R}^3\).
Standard basis vectors in \(\mathbb{R}^3\)
The standard basis vectors in \(\mathbb{R}^3\) are denoted by \(\mathbf{e}_1\), \(\mathbf{e}_2\), and \(\mathbf{e}_3\), and are defined as \[ \mathbf{e}_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \quad \mathbf{e}_2 = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, \quad \mathbf{e}_3 = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}. \]
Theorem: Resolving vectors into standard basis vectors
Any vector
\[\mathbf{u} = \begin{bmatrix} u_1 \\ u_2 \\ u_3 \end{bmatrix}\]
in \(\mathbb{R}^3\) can be expressed as a linear combination of the standard basis vectors (i.e., a sum of scalar multiples of the basis vectors) as:
\[\mathbf{u} = u_1 \mathbf{e}_1 + u_2 \mathbf{e}_2 + u_3 \mathbf{e}_3.\]