Vectors: Definitions and Properties

In VCE Specialist Mathematics, vectors are fundamental tools used to represent quantities that possess both size and orientation. This distinguishes them from scalars, which only have magnitude.

1. Introduction to Vectors

A scalar is a quantity that is fully described by its magnitude (size) and an appropriate unit. Examples include mass, time, distance, and speed.

A vector is a quantity that has both magnitude and direction. Examples include displacement, velocity, acceleration, and force.

Vector Representation and Notation

• Directed Line Segment: A vector can be represented geometrically by an arrow. The length of the arrow represents the magnitude, and the arrow-head indicates the direction.
• Notation:
  • The vector from point $A$ to point $B$ is written as $\vec{AB}$.
  • In textbooks, vectors are often printed in bold (e.g., $\mathbf{v}$).
  • In handwritten work (and exams), vectors must be written with a tilde underneath (e.g., $\tilde{v}$).
  • The position vector of a point $A$ is the vector $\vec{OA}$, where $O$ is the origin $(0,0)$ or $(0,0,0)$.

STUDY HINT: Always use the tilde notation ($\tilde{v}$) in your SACs and Exams. VCAA assessors require clear distinction between scalar variables and vector variables.

2. Component Form in Two and Three Dimensions

Vectors are most commonly expressed in terms of standard unit vectors representing the axes of the Cartesian coordinate system.

Two-Dimensional Vectors (2D)

A vector $\tilde{u}$ in 2D can be resolved into horizontal and vertical components:
$$\tilde{u} = x\tilde{i} + y\tilde{j}$$
* $\tilde{i}$: Unit vector in the positive direction of the $x$-axis.
* $\tilde{j}$: Unit vector in the positive direction of the $y$-axis.
* This can also be written as a column vector: $\begin{bmatrix} x \ y \end{bmatrix}$.

Three-Dimensional Vectors (3D)

A vector $\tilde{u}$ in 3D adds a depth component:
$$\tilde{u} = x\tilde{i} + y\tilde{j} + z\tilde{k}$$
* $\tilde{k}$: Unit vector in the positive direction of the $z$-axis.
* This can also be written as a column vector: $\begin{bmatrix} x \ y \ z \end{bmatrix}$.

VCAA FOCUS: Ensure you can switch fluently between component form ($x\tilde{i} + y\tilde{j} + z\tilde{k}$) and column vector form, as both appear frequently in exam questions.

3. Magnitude of a Vector

The magnitude (or length) of a vector is a scalar quantity denoted by $|\tilde{v}|$ or $|\vec{AB}|$. It is calculated using Pythagoras’ Theorem.

Magnitude Formulas

• In 2D: For $\tilde{v} = x\tilde{i} + y\tilde{j}$:
  $$|\tilde{v}| = \sqrt{x^2 + y^2}$$
• In 3D: For $\tilde{v} = x\tilde{i} + y\tilde{j} + z\tilde{k}$:
  $$|\tilde{v}| = \sqrt{x^2 + y^2 + z^2}$$

Properties of Magnitude

1. $|\tilde{v}| \ge 0$ (Magnitude is always non-negative).
2. $|\tilde{v}| = 0$ if and only if $\tilde{v} = \tilde{0}$ (the zero vector).
3. $|k\tilde{v}| = |k| \times |\tilde{v}|$ where $k$ is a scalar.

COMMON MISTAKE: A common error is forgetting to square negative components correctly. Remember that $(-3)^2 = 9$. The magnitude of $\tilde{v} = -3\tilde{i} + 4\tilde{j}$ is $\sqrt{(-3)^2 + 4^2} = 5$, not $\sqrt{-9 + 16}$.

4. Unit Vectors

A unit vector is a vector with a magnitude of exactly 1.

Finding a Unit Vector in a Given Direction

To find a unit vector $\hat{v}$ (pronounced “v-hat”) that points in the same direction as a non-zero vector $\tilde{v}$, divide the vector by its own magnitude:
$$\hat{v} = \frac{1}{|\tilde{v}|}\tilde{v}$$

Standard Unit Vectors

In the Cartesian system, the basis vectors are defined as:
* $\tilde{i} = \begin{bmatrix} 1 \ 0 \ 0 \end{bmatrix}$, $|\tilde{i}| = 1$
* $\tilde{j} = \begin{bmatrix} 0 \ 1 \ 0 \end{bmatrix}$, $|\tilde{j}| = 1$
* $\tilde{k} = \begin{bmatrix} 0 \ 0 \ 1 \end{bmatrix}$, $|\tilde{k}| = 1$

KEY TAKEAWAY: A unit vector represents direction only. If you need a vector of a specific length $L$ in the direction of $\tilde{v}$, the formula is $L\hat{v}$.

5. Basic Vector Operations and Properties

Equality of Vectors

Two vectors are equal if and only if they have the same magnitude and the same direction. In component form, $\tilde{u} = \tilde{v}$ if $x_1 = x_2$, $y_1 = y_2$, and $z_1 = z_2$.

Scalar Multiplication

Multiplying a vector $\tilde{v}$ by a scalar $k$:
* If $k > 0$, the direction remains the same, but the magnitude is scaled by $k$.
* If $k < 0$, the direction is reversed, and the magnitude is scaled by $|k|$.
* The vector $-\tilde{v}$ has the same magnitude as $\tilde{v}$ but the opposite direction.

Parallel Vectors

Two non-zero vectors $\tilde{u}$ and $\tilde{v}$ are parallel if one is a scalar multiple of the other:
$$\tilde{u} = k\tilde{v} \text{ for some } k \in \mathbb{R} \setminus {0}$$

Vector Addition and Subtraction

• Addition ($\tilde{u} + \tilde{v}$): Performed by adding corresponding components. Geometrically, use the “head-to-tail” rule or the parallelogram law.
• Subtraction ($\tilde{u} - \tilde{v}$): Performed by subtracting corresponding components ($\tilde{u} + (-\tilde{v})$). Geometrically, $\vec{AB} = \vec{OB} - \vec{OA}$ (Position vector of Finish minus Position vector of Start).

EXAM TIP: To prove three points $A, B,$ and $C$ are collinear (lie on the same straight line), show that the vectors $\vec{AB}$ and $\vec{BC}$ are parallel (i.e., $\vec{AB} = k\vec{BC}$) and share a common point $B$.