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Vectors and Complex Numbers

CSCA Vectors and Complex Numbers study guide organized around the publicly available CSCA syllabus. Practice Mathematics questions on aicsca.com.

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Vectors and Complex Numbers

Vectors and complex numbers are two powerful tools connecting algebra and geometry. In the CSCA exam, **Vectors** are often used to solve geometric problems involving position, angles, and distances, while **Complex Numbers** test algebraic operations and geometric interpretations (rotation and modulus).

1. Plane Vectors

A vector is a quantity with both **Magnitude** and **Direction**.

#### 1.1 Basic Operations

Let $\vec{a} = (x_1, y_1), \vec{b} = (x_2, y_2)$:

* **Addition/Subtraction**: $\vec{a} \pm \vec{b} = (x_1 \pm x_2, y_1 \pm y_2)$. Geometrically follows the **Parallelogram Law** or **Triangle Law**.

* **Modulus (Magnitude)**: $|\vec{a}| = \sqrt{x_1^2 + y_1^2}$.

#### 1.2 Dot Product (Scalar Product) — **Core Topic**

The result is a real number (scalar), bridging length and angle.

$$\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta = x_1 x_2 + y_1 y_2$$

where $\theta$ is the angle between the vectors $(0^\circ \le \theta \le 180^\circ)$.

**Key Applications**:

1. **Finding Angle**: $\cos \theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}| |\vec{b}|}$

2. **Perpendicularity**: $\vec{a} \perp \vec{b} \iff \vec{a} \cdot \vec{b} = 0 \iff x_1 x_2 + y_1 y_2 = 0$

3. **Parallelism**: $\vec{a} // \vec{b} \iff x_1 y_2 - x_2 y_1 = 0$

2. Complex Numbers

A complex number is of the form $z = a + bi$ ($a, b \in \mathbb{R}, i^2 = -1$). It can be viewed as a point $Z(a, b)$ or a vector $\vec{OZ}$ on the plane.

#### 2.1 Geometry and Modulus

* **Complex Plane**: x-axis is the Real Axis, y-axis is the Imaginary Axis.

* **Modulus**: $|z| = \sqrt{a^2 + b^2}$, representing the distance from the origin.

* **Conjugate**: $\bar{z} = a - bi$, symmetric to $z$ about the Real Axis.

#### 2.2 Operations and Geometric Transformations

* **Arithmetic**: Follows algebraic rules, noting $i^2 = -1$.

* Division trick: Rationalize the denominator (multiply top and bottom by the conjugate of the denominator).

* **Rotation**:

* Multiply by $i$: Rotate $90^\circ$ counter-clockwise.

* Multiply by $-1$: Rotate $180^\circ$ (Central symmetry).

3. Practice Examples

**Example 1**: Given vectors $\vec{a} = (1, 2), \vec{b} = (-3, k)$, if $\vec{a} \perp \vec{b}$, find $k$.

**Solution**:

Since they are perpendicular, $\vec{a} \cdot \vec{b} = 0$.

$1 \times (-3) + 2 \times k = 0 \Rightarrow -3 + 2k = 0 \Rightarrow k = 1.5$.

**Example 2**: If $z = \frac{2i}{1-i}$, find $|z|$.

**Solution**:

Simplify: $z = \frac{2i(1+i)}{(1-i)(1+i)} = \frac{2i - 2}{1 - (-1)} = \frac{-2 + 2i}{2} = -1 + i$.

Modulus: $|z| = \sqrt{(-1)^2 + 1^2} = \sqrt{2}$.

4. Study Tips

* **Vectors**: See "Perpendicular" $\to$ think "Dot product is 0"; see "Parallel" $\to$ think "Cross product of coords is 0".

* **Complex Numbers**: Master rationalizing denominators; understand $|z_1 - z_2|$ as distance between two points.