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URL: https://nakafa.com/en/subjects/mathematics/complex-number/scalar-multiplication-complex-numbers
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/material/lesson/mathematics/complex-number/scalar-multiplication-complex-numbers/en.mdx

Explore scalar multiplication effects on complex vectors with interactive visualizations showing stretching, shrinking, and direction reversal.

---

## What is Scalar Multiplication?

Scalar multiplication involves multiplying a complex number by a real number (a scalar).

Component: LineEquation
Props:
- title: Visualization of Scalar Multiplication
- description: Notice how the vector $$z = 1+2i$$ changes when
multiplied by the scalar $$c=2$$ and{" "}
$$c=-0.5$$.
  Visible text: Notice how the vector changes when
multiplied by the scalar and{" "}
.
- cameraPosition: [0, 0, 12]
- showZAxis: false
- data: [
// Original vector z
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: 1, y: 2, z: 0 },
],
color: getColor("SKY"),
labels: [{ text: "z", at: 1, offset: [0.5, 0.5, 0] }],
cone: { position: "end" },
},
// Vector 2z
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: 2, y: 4, z: 0 }, // 2 * (1+2i) = 2+4i
],
color: getColor("EMERALD"),
labels: [{ text: "2z", at: 1, offset: [0.5, 0.5, 0] }],
cone: { position: "end" },
},
// Vector -0.5z
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: -0.5, y: -1, z: 0 }, // -0.5 * (1+2i) = -0.5 - i
],
color: getColor("ROSE"),
labels: [{ text: "-0.5z", at: 1, offset: [-0.5, -0.5, 0] }],
cone: { position: "end" },
},
]

From the visualization above, we can see:

- Multiplying $$z$$ by a scalar $$c > 1$$ (like $$2$$) will **stretch** the vector $$z$$ in the same direction.
- Multiplying $$z$$ by a scalar $$0 < c < 1$$ will **shrink** the vector $$z$$ in the same direction.
- Multiplying $$z$$ by a scalar $$c < 0$$ (like $$-0.5$$) will **reverse the direction** of the vector $$z$$ (by $$180^\circ$$) and change its length according to the value of $$|c|$$.

Visible text: - Multiplying by a scalar (like ) will **stretch** the vector in the same direction.
- Multiplying by a scalar will **shrink** the vector in the same direction.
- Multiplying by a scalar (like ) will **reverse the direction** of the vector (by ) and change its length according to the value of .

## Mathematical Definition

If $$z = x + iy$$ is a complex number and $$c$$ is a scalar (a real number), then their scalar multiplication is:

Visible text: If is a complex number and is a scalar (a real number), then their scalar multiplication is:

```math
cz = c(x + iy) = (cx) + i(cy)
```

This means we simply multiply the scalar $$c$$ by the real part ($$x$$) and the imaginary part ($$y$$) separately.

Visible text: This means we simply multiply the scalar by the real part () and the imaginary part () separately.

### Calculation Examples

If $$z = -3 + 4i$$ and $$c = 3$$, then:

Visible text: If and , then:

```math
cz = 3(-3 + 4i) = 3(-3) + i(3 \times 4) = -9 + 12i
```

If $$z = 5 - i$$ and $$c = -2$$, then:

Visible text: If and , then:

```math
cz = -2(5 - i) = -2(5) + i(-2 \times -1) = -10 + 2i
```

## Visualization Examples

Let's look at a few more examples to clarify the effect of scalar multiplication.

### Scalar Greater than One

Component: LineEquation
Props:
- title: Multiplication by Scalar $$c = 1.5$$
  Visible text: Multiplication by Scalar
- description: Vector $$z = -2 + i$$ is stretched in the same direction
when multiplied by $$c = 1.5$$, becoming{" "}
$$1.5z = -3 + 1.5i$$.
  Visible text: Vector is stretched in the same direction
when multiplied by , becoming{" "}
.
- cameraPosition: [0, 0, 8]
- showZAxis: false
- data: [
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: -2, y: 1, z: 0 },
],
color: getColor("VIOLET"),
labels: [{ text: "z", at: 1, offset: [0.5, 0.5, 0] }],
cone: { position: "end" },
},
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: -3, y: 1.5, z: 0 }, // 1.5 * (-2+i)
],
color: getColor("LIME"),
labels: [{ text: "1.5z", at: 1, offset: [-0.5, 0.5, 0] }],
cone: { position: "end" },
},
]

### Scalar between Zero and One

Component: LineEquation
Props:
- title: Multiplication by Scalar $$c = 0.75$$
  Visible text: Multiplication by Scalar
- description: Vector $$z = 3 - 2i$$ is shrunk in the same direction
when multiplied by $$c = 0.75$$, becoming{" "}
$$0.75z = 2.25 - 1.5i$$.
  Visible text: Vector is shrunk in the same direction
when multiplied by , becoming{" "}
.
- cameraPosition: [0, 0, 8]
- showZAxis: false
- data: [
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: 3, y: -2, z: 0 },
],
color: getColor("ORANGE"),
labels: [{ text: "z", at: 1, offset: [-0.5, -0.5, 0] }],
cone: { position: "end" },
},
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: 2.25, y: -1.5, z: 0 }, // 0.75 * (3-2i)
],
color: getColor("TEAL"),
labels: [{ text: "0.75z", at: 1, offset: [0.5, 1, 0] }],
cone: { position: "end" },
},
]

### Negative Scalar

Multiplication by $$-1$$ yields the additive inverse (negative) of the complex number.

Visible text: Multiplication by yields the additive inverse (negative) of the complex number.

Component: LineEquation
Props:
- title: Multiplication by Scalar $$c = -1$$ (Additive Inverse)
  Visible text: Multiplication by Scalar (Additive Inverse)
- description: Vector $$z = -1 - 3i$$ reverses direction ($$180^\circ$$) when
multiplied by $$c = -1$$, becoming{" "}
$$-z = 1 + 3i$$.
  Visible text: Vector reverses direction () when
multiplied by , becoming{" "}
.
- cameraPosition: [0, 0, 10]
- showZAxis: false
- data: [
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: -1, y: -3, z: 0 },
],
color: getColor("FUCHSIA"),
labels: [{ text: "z", at: 1, offset: [-0.5, -0.5, 0] }],
cone: { position: "end" },
},
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: 1, y: 3, z: 0 }, // -1 * (-1-3i)
],
color: getColor("YELLOW"),
labels: [{ text: "-z", at: 1, offset: [0.5, 0.5, 0] }],
cone: { position: "end" },
},
]