Vectors in Two and Three Dimensions

A vector is a quantity that possesses both magnitude and direction. In VCE Specialist Mathematics, vectors are used to represent physical quantities such as displacement, velocity, and force, and to solve complex geometric problems in 2D and 3D space.

1. Vector Representation and Notation

Vectors can be represented geometrically as directed line segments or algebraically using components.

Component Form

• 2D Vectors: $\mathbf{r} = x\mathbf{i} + y\mathbf{j}$
• 3D Vectors: $\mathbf{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k}$

Where $\mathbf{i}, \mathbf{j}, \mathbf{k}$ are unit vectors in the directions of the positive $x, y,$ and $z$ axes respectively.
* $\mathbf{i} = \begin{bmatrix} 1 \ 0 \ 0 \end{bmatrix}$, $\mathbf{j} = \begin{bmatrix} 0 \ 1 \ 0 \end{bmatrix}$, $\mathbf{k} = \begin{bmatrix} 0 \ 0 \ 1 \end{bmatrix}$

Magnitude

The magnitude (length) of a vector $\mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} + a_3\mathbf{k}$ is given by:
$$|\mathbf{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2}$$

Unit Vectors

A unit vector is a vector with a magnitude of 1. To find the unit vector $\hat{\mathbf{a}}$ in the direction of $\mathbf{a}$:
$$\hat{\mathbf{a}} = \frac{1}{|\mathbf{a}|}\mathbf{a}$$

KEY TAKEAWAY: A vector is defined by its components. To convert any vector into a unit vector, divide the vector by its own magnitude. This is essential for finding vector resolutes.

2. Vector Algebra

Basic operations allow for the manipulation of vectors in space.

Parallel Vectors

Two non-zero vectors $\mathbf{a}$ and $\mathbf{b}$ are parallel if one is a scalar multiple of the other:
$$\mathbf{a} = k\mathbf{b} \text{ for some } k \in \mathbb{R} \setminus {0}$$
* If $k > 0$, they are in the same direction.
* If $k < 0$, they are in opposite directions.

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

3. Linear Dependence and Independence

A set of vectors ${\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_n}$ is linearly independent if the equation:
$$k_1\mathbf{v}_1 + k_2\mathbf{v}_2 + \dots + k_n\mathbf{v}_n = \mathbf{0}$$
has only the trivial solution $k_1 = k_2 = \dots = k_n = 0$.

• In 2D: Two vectors are linearly independent if they are not parallel.
• In 3D: Three vectors are linearly independent if they do not lie in the same plane (non-coplanar).

VCAA FOCUS: If a vector $\mathbf{r}$ can be expressed as $\mathbf{r} = m\mathbf{a} + n\mathbf{b}$, then $\mathbf{r}$ is linearly dependent on $\mathbf{a}$ and $\mathbf{b}$, meaning it lies in the same plane as $\mathbf{a}$ and $\mathbf{b}$.

4. The Scalar (Dot) Product

The scalar product results in a scalar value and is used to find angles and determine perpendicularity.

Algebraic Definition

For $\mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} + a_3\mathbf{k}$ and $\mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j} + b_3\mathbf{k}$:
$$\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3$$

Geometric Definition

$$\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}||\mathbf{b}| \cos(\theta)$$
where $\theta$ is the angle between the two vectors (\$0 \le \theta \le \pi$).

Properties

1. Perpendicularity: $\mathbf{a} \cdot \mathbf{b} = 0 \iff \mathbf{a} \perp \mathbf{b}$ (for non-zero vectors).
2. Magnitude squared: $\mathbf{a} \cdot \mathbf{a} = |\mathbf{a}|^2$.
3. Angle between vectors: $\cos(\theta) = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}||\mathbf{b}|}$.

COMMON MISTAKE: Students often forget that the dot product results in a number (scalar), not a vector. Writing $\mathbf{a} \cdot \mathbf{b} = 5\mathbf{i}$ is a fundamental notation error.

5. Vector and Scalar Resolutes

Resolving a vector $\mathbf{a}$ involves breaking it into two components: one parallel to $\mathbf{b}$ and one perpendicular to $\mathbf{b}$.

Scalar Resolute

The scalar resolute of $\mathbf{a}$ in the direction of $\mathbf{b}$ is the “length” of the projection:
$$\text{scalar resolute} = \mathbf{a} \cdot \hat{\mathbf{b}} = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|}$$

Vector Resolute Parallel to b

The component of $\mathbf{a}$ that lies in the direction of $\mathbf{b}$:
$$\mathbf{a}_{||} = (\mathbf{a} \cdot \hat{\mathbf{b}})\hat{\mathbf{b}} = \left( \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|^2} \right) \mathbf{b}$$

Vector Resolute Perpendicular to b

The component of $\mathbf{a}$ that is orthogonal to $\mathbf{b}$:
$$\mathbf{a}{\perp} = \mathbf{a} - \mathbf{a}{||}$$

STUDY HINT: Always check that $\mathbf{a}{||} + \mathbf{a}{\perp} = \mathbf{a}$ and $\mathbf{a}{||} \cdot \mathbf{a}{\perp} = 0$ to verify your calculations.

6. The Vector (Cross) Product

The vector product $\mathbf{a} \times \mathbf{b}$ results in a vector that is perpendicular to both $\mathbf{a}$ and $\mathbf{b}$.

Calculation (Determinant Method)

For $\mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} + a_3\mathbf{k}$ and $\mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j} + b_3\mathbf{k}$:
$$\mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \end{vmatrix} = (a_2b_3 - a_3b_2)\mathbf{i} - (a_1b_3 - a_3b_1)\mathbf{j} + (a_1b_2 - a_2b_1)\mathbf{k}$$

Magnitude and Geometric Meaning

• $|\mathbf{a} \times \mathbf{b}| = |\mathbf{a}||\mathbf{b}| \sin(\theta)$
• The magnitude $|\mathbf{a} \times \mathbf{b}|$ is the area of the parallelogram formed by $\mathbf{a}$ and $\mathbf{b}$.
• The area of a triangle formed by $\mathbf{a}$ and $\mathbf{b}$ is $\frac{1}{2}|\mathbf{a} \times \mathbf{b}|$.

REMEMBER: The cross product is anti-commutative: $\mathbf{a} \times \mathbf{b} = -(\mathbf{b} \times \mathbf{a})$. The direction is determined by the right-hand grip rule.

7. Geometric Applications

Vectors are used to prove geometric theorems.

Common Proof Techniques:

1. Midpoint of a line segment: $\vec{M} = \frac{1}{2}(\vec{A} + \vec{B})$.
2. Dot Product for Orthogonality: Use $\mathbf{a} \cdot \mathbf{b} = 0$ to prove right angles (e.g., altitudes of a triangle, or that the diagonals of a rhombus are perpendicular).
3. Linear Combinations: Use linear independence to find ratios in which lines divide each other.

Key Geometric Results:

• The Diagonals of a Parallelogram: Bisect each other.
• The Medians of a Triangle: Are concurrent at the centroid, which divides each median in the ratio \$2:1$.
• Angle in a Semicircle: Use vectors from the center to the circumference to prove the angle is $90^\circ$ ($\mathbf{a} \cdot \mathbf{b} = 0$).

APPLICATION: In kinematics, if $\mathbf{r}(t)$ is the position vector of a particle, then $\mathbf{v}(t) = \frac{d\mathbf{r}}{dt}$ is the velocity vector and $\mathbf{a}(t) = \frac{d\mathbf{v}}{dt}$ is the acceleration vector. The dot product is often used to find when velocity is perpendicular to acceleration.