Vector Operations

Vector operations lay the foundation for understanding direction and magnitude. Through addition, subtraction, and scalar multiplication, we combine and scale vectors—key steps in navigating higher-dimensional spaces. ➡️

Vector Addition

Adding vectors places them head-to-tail, and the result is the diagonal of the parallelogram they form.

\[ \mathbf{u} + \mathbf{v} = \begin{bmatrix} u_1 \\ u_2 \end{bmatrix} + \begin{bmatrix} v_1 \\ v_2 \end{bmatrix} = \begin{bmatrix} u_1 + v_1 \\ u_2 + v_2 \end{bmatrix} \]

viewof u1_add = Inputs.range([-8, 8], {
  step: 0.5, 
  value: 3, 
  label: tex`u_1`, 
  width: 180
})

viewof u2_add = Inputs.range([-8, 8], {
  step: 0.5, 
  value: 2, 
  label: tex`u_2`, 
  width: 180
})

// Vector v components for addition
viewof v1_add = Inputs.range([-8, 8], {
  step: 0.5, 
  value: -2, 
  label: tex`v_1`, 
  width: 180
})

viewof v2_add = Inputs.range([-8, 8], {
  step: 0.5, 
  value: 4, 
  label: tex`v_2`, 
  width: 180
})

// Display vectors and sum
html`<div style="text-align: center; font-size: 1.2em; margin: 1.5em 0;">
  <p>
    ${tex`\mathbf{u} = \begin{bmatrix} ${u1_add} \\ ${u2_add} \end{bmatrix}, \quad \mathbf{v} = \begin{bmatrix} ${v1_add} \\ ${v2_add} \end{bmatrix}, \quad \mathbf{u} + \mathbf{v} = \begin{bmatrix} ${u1_add + v1_add} \\ ${u2_add + v2_add} \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: [
    Plot.ruleY([0], {stroke: "#ddd"}),
    Plot.ruleX([0], {stroke: "#ddd"}),
    
    // Vector u (blue)
    Plot.arrow([{x1: 0, y1: 0, x2: u1_add, y2: u2_add}], {
      x1: "x1", y1: "y1", x2: "x2", y2: "y2",
      stroke: "steelblue", strokeWidth: 3, fill: "steelblue"
    }),
    
    // Vector v (red)
    Plot.arrow([{x1: 0, y1: 0, x2: v1_add, y2: v2_add}], {
      x1: "x1", y1: "y1", x2: "x2", y2: "y2",
      stroke: "crimson", strokeWidth: 3, fill: "crimson"
    }),
    
    // Vector sum u + v (green)
    Plot.arrow([{x1: 0, y1: 0, x2: u1_add + v1_add, y2: u2_add + v2_add}], {
      x1: "x1", y1: "y1", x2: "x2", y2: "y2",
      stroke: "green", strokeWidth: 3, fill: "green"
    }),
    
    // Parallelogram lines
    Plot.line([{x: u1_add, y: u2_add}, {x: u1_add + v1_add, y: u2_add + v2_add}], {
      x: "x", y: "y", stroke: "#999", strokeWidth: 1, strokeDasharray: "5,5"
    }),
    Plot.line([{x: v1_add, y: v2_add}, {x: u1_add + v1_add, y: u2_add + v2_add}], {
      x: "x", y: "y", stroke: "#999", strokeWidth: 1, strokeDasharray: "5,5"
    }),
    
    // Labels
    Plot.text([
      {x: u1_add/2, y: u2_add/2 + 0.5, text: "u"},
      {x: v1_add/2, y: v2_add/2 - 0.5, text: "v"},
      {x: (u1_add + v1_add)/2, y: (u2_add + v2_add)/2 + 0.8, text: "u+v"}
    ], {
      x: "x", y: "y", text: "text", fontSize: 14, fontWeight: "bold"
    })
  ]
})

Vector Subtraction

Subtracting vectors gives the vector that points from the tip of (\(\mathbf{v}\)) to the tip of (\(\mathbf{u}\)).

\[ \mathbf{u} - \mathbf{v} = \begin{bmatrix} u_1 \\ u_2 \end{bmatrix} - \begin{bmatrix} v_1 \\ v_2 \end{bmatrix} = \begin{bmatrix} u_1 - v_1 \\ u_2 - v_2 \end{bmatrix} \]

viewof u1_sub = Inputs.range([-8, 8], {
  step: 0.5, 
  value: 4, 
  label: tex`u_1`, 
  width: 180
})

viewof u2_sub = Inputs.range([-8, 8], {
  step: 0.5, 
  value: 3, 
  label: tex`u_2`, 
  width: 180
})

// Vector v components for subtraction
viewof v1_sub = Inputs.range([-8, 8], {
  step: 0.5, 
  value: 1, 
  label: tex`v_1`, 
  width: 180
})

viewof v2_sub = Inputs.range([-8, 8], {
  step: 0.5, 
  value: 2, 
  label: tex`v_2`, 
  width: 180
})

// Display vectors and difference
html`<div style="text-align: center; font-size: 1.2em; margin: 1.5em 0;">
  <p>
    ${tex`\mathbf{u} = \begin{bmatrix} ${u1_sub} \\ ${u2_sub} \end{bmatrix}, \quad \mathbf{v} = \begin{bmatrix} ${v1_sub} \\ ${v2_sub} \end{bmatrix}, \quad \mathbf{u} - \mathbf{v} = \begin{bmatrix} ${u1_sub - v1_sub} \\ ${u2_sub - v2_sub} \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: [
    Plot.ruleY([0], {stroke: "#ddd"}),
    Plot.ruleX([0], {stroke: "#ddd"}),
    
    // Vector u (blue)
    Plot.arrow([{x1: 0, y1: 0, x2: u1_sub, y2: u2_sub}], {
      x1: "x1", y1: "y1", x2: "x2", y2: "y2",
      stroke: "steelblue", strokeWidth: 3, fill: "steelblue"
    }),
    
    // Vector v (red)
    Plot.arrow([{x1: 0, y1: 0, x2: v1_sub, y2: v2_sub}], {
      x1: "x1", y1: "y1", x2: "x2", y2: "y2",
      stroke: "crimson", strokeWidth: 3, fill: "crimson"
    }),
    
    // Vector -v (red dashed)
    Plot.arrow([{x1: 0, y1: 0, x2: -v1_sub, y2: -v2_sub}], {
      x1: "x1", y1: "y1", x2: "x2", y2: "y2",
      stroke: "crimson", strokeWidth: 2, fill: "crimson", 
      strokeDasharray: "5,5", fillOpacity: 0.6
    }),
    
    // Vector difference u - v (purple)
    Plot.arrow([{x1: 0, y1: 0, x2: u1_sub - v1_sub, y2: u2_sub - v2_sub}], {
      x1: "x1", y1: "y1", x2: "x2", y2: "y2",
      stroke: "purple", strokeWidth: 3, fill: "purple"
    }),
    
    // Vector from v to u (alternative representation)
    Plot.arrow([{x1: v1_sub, y1: v2_sub, x2: u1_sub, y2: u2_sub}], {
      x1: "x1", y1: "y1", x2: "x2", y2: "y2",
      stroke: "#ff6600", strokeWidth: 2, fill: "#ff6600", 
      strokeDasharray: "3,3"
    }),
    
    // Labels
    Plot.text([
      {x: u1_sub/2, y: u2_sub/2 + 0.5, text: "u"},
      {x: v1_sub/2, y: v2_sub/2 - 0.5, text: "v"},
      {x: -v1_sub/2, y: -v2_sub/2 + 0.5, text: "-v"},
      {x: (u1_sub - v1_sub)/2, y: (u2_sub - v2_sub)/2 + 0.8, text: "u-v"}
    ], {
      x: "x", y: "y", text: "text", fontSize: 14, fontWeight: "bold"
    })
  ]
})

Scalar Multiplication

Multiplying a vector by a scalar stretches or shrinks its length, and reverses its direction if the scalar is negative.

\[ k\mathbf{u} = k \begin{bmatrix} u_1 \\ u_2 \end{bmatrix} = \begin{bmatrix} k u_1 \\ k u_2 \end{bmatrix} \]

viewof scalar = Inputs.range([-3, 3], {
  step: 0.1, 
  value: 1.5, 
  label: tex`k`, 
  width: 200
})

// Vector u components for scalar multiplication
viewof u1_scalar = Inputs.range([-6, 6], {
  step: 0.5, 
  value: 3, 
  label: tex`u_1`, 
  width: 180
})

viewof u2_scalar = Inputs.range([-6, 6], {
  step: 0.5, 
  value: 2, 
  label: tex`u_2`, 
  width: 180
})

// Display vectors and scalar product
html`<div style="text-align: center; font-size: 1.2em; margin: 1.5em 0;">
  <p>
    ${tex`k = ${scalar.toFixed(1)}, \quad \mathbf{u} = \begin{bmatrix} ${u1_scalar} \\ ${u2_scalar} \end{bmatrix}, \quad k\mathbf{u} = \begin{bmatrix} ${(scalar * u1_scalar).toFixed(1)} \\ ${(scalar * u2_scalar).toFixed(1)} \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: [
    Plot.ruleY([0], {stroke: "#ddd"}),
    Plot.ruleX([0], {stroke: "#ddd"}),
    
    // Original vector u (blue)
    Plot.arrow([{x1: 0, y1: 0, x2: u1_scalar, y2: u2_scalar}], {
      x1: "x1", y1: "y1", x2: "x2", y2: "y2",
      stroke: "steelblue", strokeWidth: 3, fill: "steelblue"
    }),
    
    // Scaled vector ku (orange)
    Plot.arrow([{x1: 0, y1: 0, x2: scalar * u1_scalar, y2: scalar * u2_scalar}], {
      x1: "x1", y1: "y1", x2: "x2", y2: "y2",
      stroke: "orange", strokeWidth: 3, fill: "orange"
    }),
    
    // Unit vector in same direction (if scalar != 0)
    scalar !== 0 ? Plot.arrow([{
      x1: 0, y1: 0, 
      x2: u1_scalar / Math.sqrt(u1_scalar*u1_scalar + u2_scalar*u2_scalar), 
      y2: u2_scalar / Math.sqrt(u1_scalar*u1_scalar + u2_scalar*u2_scalar)
    }], {
      x1: "x1", y1: "y1", x2: "x2", y2: "y2",
      stroke: "#666", strokeWidth: 1, fill: "#666", 
      strokeDasharray: "2,2"
    }) : null,
    
    // Labels
    Plot.text([
      {x: u1_scalar/2, y: u2_scalar/2 + 0.5, text: "u"},
      {x: (scalar * u1_scalar)/2, y: (scalar * u2_scalar)/2 - 0.8, text: `${scalar.toFixed(1)}u`}
    ], {
      x: "x", y: "y", text: "text", fontSize: 14, fontWeight: "bold"
    })
  ]
})