Vectors

From direction and magnitude to data points and gradients—vectors are the foundation of linear algebra. They shape how we navigate, calculate, and represent multidimensional spaces. 🧭

Vector Notation

A vector is simply a list of numbers that are ordered in \(\mathbb{R}^n\).

\(n\) is a positive integer indicating the number of entries.

In general, a vector \(\mathbf{v} \in \mathbb{R}^n\) can be written as:

\[ \mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix} \]

For example, in \(\mathbb{R}^2\):

\[ \mathbf{u} = \begin{bmatrix} 0 \\ 1 \end{bmatrix} \hspace{0.4cm} \mathbf{v} = \begin{bmatrix} 1 \\ 1 \end{bmatrix} \]

Suppose we have the following matrix:

\[ A = \begin{bmatrix} 2 & -1 & 1 \\ 1 & 0 & 0 \\ 0 & 0 & 4 \end{bmatrix} \]

One standard derivation of vectors from matrices are column vectors.

For our example, the first column vector is:

\[ \left[ \begin{array}{c c c} \color{red}{\boxed{2}} & -1 & 1 \\ \color{red}{\boxed{1}} & 0 & 0 \\ \color{red}{\boxed{0}} & 0 & 4 \end{array} \right] \]

\[ \mathbf{a}_1= \begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix} \]

Another standard derivation of vectors from matrices are row vectors.

For our example, the first row vector is:

\[ \left[ \begin{array}{c c c} \color{red}{\boxed{2}} & \color{red}{\boxed{-1}} & \color{red}{\boxed{1}} \\ 1 & 0 & 0 \\ 0 & 0 & 4 \end{array} \right] \]

\[ \mathbf{a}^T_1= \begin{bmatrix} 2 & -1 & 1 \end{bmatrix} \]

viewof u1 = Inputs.range([-10, 10], {
  step: 0.5, 
  value: 3, 
  label: tex`u_1`, 
  width: 200
})

viewof u2 = Inputs.range([-10, 10], {
  step: 0.5, 
  value: 2, 
  label: tex`u_2`, 
  width: 200
})

// Vector v components
viewof v1 = Inputs.range([-10, 10], {
  step: 0.5, 
  value: -2, 
  label: tex`v_1`, 
  width: 200
})

viewof v2 = Inputs.range([-10, 10], {
  step: 0.5, 
  value: 4, 
  label: tex`v_2`, 
  width: 200
})

// Display vectors
html`<div style="text-align: center; font-size: 1.3em; margin: 2em 0;">
  <p style="margin-top: 1em;">
    ${tex`\mathbf{u} = \begin{bmatrix} ${u1} \\ ${u2} \end{bmatrix} \quad \mathbf{v} = \begin{bmatrix} ${v1} \\ ${v2} \end{bmatrix}`}
  </p>
</div>`

Plot.plot({
  style: "overflow: visible; display: block; margin: 0 auto;",
  width: 600,
  height: 500,
  grid: true,
  x: {label: "x_1", domain: [-12, 12]},
  y: {label: "x_2", domain: [-12, 12]},
  marks: [
    // Grid axes
    Plot.ruleY([0], {stroke: "#ccc"}),
    Plot.ruleX([0], {stroke: "#ccc"}),
    
    // Vector u (blue arrow)
    Plot.arrow([{x1: 0, y1: 0, x2: u1, y2: u2}], {
      x1: "x1", y1: "y1", x2: "x2", y2: "y2",
      stroke: "steelblue",
      strokeWidth: 3,
      fill: "steelblue"
    }),
    
    // Vector v (red arrow)
    Plot.arrow([{x1: 0, y1: 0, x2: v1, y2: v2}], {
      x1: "x1", y1: "y1", x2: "x2", y2: "y2",
      stroke: "crimson",
      strokeWidth: 3,
      fill: "crimson"
    }),
    
    // Vector endpoints
    Plot.dot([
      {x: u1, y: u2, color: "u"},
      {x: v1, y: v2, color: "v"}
    ], {
      x: "x", y: "y",
      r: 4,
      fill: d => d.color === "u" ? "steelblue" : "crimson",
      stroke: "white",
      strokeWidth: 1
    }),
    
    // Labels
    Plot.text([
      {x: u1/2, y: u2/2 + 0.5, text: "u"},
      {x: v1/2, y: v2/2 - 0.5, text: "v"}
    ], {
      x: "x", y: "y", text: "text",
      fontSize: 14,
      fontWeight: "bold"
    })
  ]
})