Vectors in Geometry and Mechanics

Vectors are quantities that possess both magnitude and direction. In VCE Specialist Mathematics, vectors are applied to solve complex problems in three-dimensional geometry and the physical behavior of objects under the influence of forces.

1. Vector Fundamentals in 3D

A vector in three dimensions is represented using the unit vectors \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\), which correspond to the \(x\), \(y\), and \(z\) axes respectively.

Component Form

A vector \(\mathbf{a}\) can be written as:
\$\(\mathbf{a} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} \text{ or } \mathbf{a} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}\)\$

Magnitude and Unit Vectors

• Magnitude: The length of vector \(\mathbf{a}\) is given by \(|\mathbf{a}| = \sqrt{x^2 + y^2 + z^2}\).
• Unit Vector: A vector with a magnitude of 1 in the direction of \(\mathbf{a}\), denoted as \(\hat{\mathbf{a}}\):
  \$\(\hat{\mathbf{a}} = \frac{1}{|\mathbf{a}|}\mathbf{a}\)\$

Parallel and Linearly Independent Vectors

• Parallel Vectors: Two vectors \(\mathbf{a}\) and \(\mathbf{b}\) are parallel if \(\mathbf{a} = k\mathbf{b}\) for some scalar \(k \in \mathbb{R} \setminus \{0\}\).
• Linear Independence: Two vectors are linearly independent if they are not parallel. In 2D, any vector can be expressed as a linear combination of two non-parallel vectors. In 3D, three vectors are linearly independent if they do not lie in the same plane (non-coplanar).

EXAM TIP: When proving three points \(A, B,\) and \(C\) are collinear, show that \(\vec{AB} = k\vec{BC}\). This confirms they are parallel and share a common point.

2. The Scalar (Dot) Product

The scalar product is a fundamental tool for finding angles and determining perpendicularity.

Definition

For two vectors \(\mathbf{a}\) and \(\mathbf{b}\) with an angle \(\theta\) between them:
\$\(\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}||\mathbf{b}|\cos(\theta)\)\$
In component form:
\$\(\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3\)\$

Key Properties

COMMON MISTAKE: Students often forget that the angle \(\theta\) between two vectors must be calculated when the vectors are placed “tail-to-tail.”

3. Vector Resolutes

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

• Scalar Resolute of \(\mathbf{a}\) in the direction of \(\mathbf{b}\):
  \$\(s = \mathbf{a} \cdot \hat{\mathbf{b}} = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|}\)\$
• Vector Resolute of \(\mathbf{a}\) parallel to \(\mathbf{b}\) (\(\mathbf{a}_{\|}\)):
  \$\(\mathbf{a}_{\|} = (\mathbf{a} \cdot \hat{\mathbf{b}})\hat{\mathbf{b}} = \left( \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|^2} \right) \mathbf{b}\)\$
• Vector Resolute of \(\mathbf{a}\) perpendicular to \(\mathbf{b}\) (\(\mathbf{a}_{\perp}\)):
  \$\(\mathbf{a}_{\perp} = \mathbf{a} - \mathbf{a}_{\|}\)\$

KEY TAKEAWAY: The vector resolute \(\mathbf{a}_{\|}\) is a vector (it has \(\mathbf{i}, \mathbf{j}, \mathbf{k}\) components), whereas the scalar resolute is simply a real number representing the magnitude of that projection (with sign indicating direction).

4. Geometric Proofs using Vectors

Vectors can be used to prove geometric theorems in both 2D and 3D.

Common Strategies

1. Midpoints: If \(M\) is the midpoint of \(AB\), then \(\vec{OM} = \frac{1}{2}(\vec{OA} + \vec{OB})\).
2. Dot Product for Orthogonality: To prove two lines are perpendicular (e.g., diagonals of a rhombus), show their vector dot product is zero.
3. Linear Combination: To show a point lies on a line segment, express its position vector in terms of the endpoints.

Example Theorems to Prove:
* The diagonals of a rhombus bisect each other at right angles.
* The angle subtended by a diameter in a circle is a right angle.
* The medians of a triangle are concurrent.

VCAA FOCUS: Proof-based questions often require clear communication. Define your vectors (e.g., “Let \(\vec{OA} = \mathbf{a}\)”) at the start and clearly state the geometric implication of your final vector result (e.g., “Since \(\mathbf{a} \cdot \mathbf{b} = 0\), \(OA \perp OB\)”).

5. Mechanics: Forces and Equilibrium

In mechanics, forces are treated as vectors. The behavior of a particle depends on the resultant force (\(\sum \mathbf{F}\)).

Equilibrium

A particle is in equilibrium if the vector sum of all forces acting on it is zero:
\$\(\mathbf{R} = \mathbf{F}_1 + \mathbf{F}_2 + \dots + \mathbf{F}_n = \mathbf{0}\)\$
In component form, this means:
\$\(\sum F_x = 0 \quad \text{and} \quad \sum F_y = 0 \quad (\text{and } \sum F_z = 0 \text{ in 3D})\)\$

Lami’s Theorem

For three coplanar forces \(\mathbf{F}_1, \mathbf{F}_2, \mathbf{F}_3\) acting at a point in equilibrium, if \(\alpha, \beta, \gamma\) are the angles opposite the forces:
\$\(\frac{|\mathbf{F}_1|}{\sin(\alpha)} = \frac{|\mathbf{F}_2|}{\sin(\beta)} = \frac{|\mathbf{F}_3|}{\sin(\gamma)}\)\$

Specific Forces

• Weight (\(W\)): \(W = mg\) (acting downwards).
• Normal Reaction (\(N\)): Acts perpendicular to the surface of contact.
• Friction (\(F_f\)): Acts parallel to the surface, opposing motion.
• Tension (\(T\)): Acts along a string or rod, pulling away from the object.

STUDY HINT: Always draw a Free Body Diagram (FBD). Resolving forces into components (usually parallel and perpendicular to an inclined plane) is almost always the first step in solving mechanics problems.

6. Motion in a Plane

When an object moves in a 2D plane, its position, velocity, and acceleration are vector functions of time \(t\).

Vector Kinematics

1. Position: \(\mathbf{r}(t) = x(t)\mathbf{i} + y(t)\mathbf{j}\)
2. Velocity: \(\mathbf{v}(t) = \frac{d\mathbf{r}}{dt} = \dot{x}(t)\mathbf{i} + \dot{y}(t)\mathbf{j}\)
3. Acceleration: \(\mathbf{a}(t) = \frac{d\mathbf{v}}{dt} = \ddot{x}(t)\mathbf{i} + \ddot{y}(t)\mathbf{j}\)

Constant Acceleration Equations

If acceleration \(\mathbf{a}\) is constant:
* \(\mathbf{v} = \mathbf{u} + \mathbf{a}t\)
* \(\mathbf{r} = \mathbf{u}t + \frac{1}{2}\mathbf{a}t^2 + \mathbf{r}_0\)
* Average velocity \(= \frac{\Delta \mathbf{r}}{\Delta t}\)

REMEMBER: Displacement is the change in position: \(\Delta \mathbf{r} = \mathbf{r}(t_2) - \mathbf{r}(t_1)\). Distance is the integral of speed: \(\int_{t_1}^{t_2} |\mathbf{v}(t)| \, dt\).