Vectors Key Terms: A Comprehensive Glossary

Introduction

Moving from 2D coordinate geometry into 3D space requires an entirely new algebraic toolkit. In HSC Mathematics Extension 2, Vectors are the language of this 3D space. This glossary provides the definitive mathematical definitions and illustrative examples for the core terminology used throughout the Further Work with Vectors syllabus.

Executive Summary

This guide serves as a quick-reference dictionary for:

Equations of Lines: Vector equation, Cartesian equation, gradient-intercept form, parameter.
Line Interactions: Parallel, collinear, skew lines.
Vector Operations: Scalar product, commutative law, distributive law, Cauchy-Schwarz inequality.
Vector Properties: Direction vector, position vector, scalar.

What is this about?

When a question asks you to "find the vector equation of the line", providing a Cartesian equation will score you zero marks. You must know the exact difference between a "position vector" (which anchors a point) and a "direction vector" (which shows where the line is going). This glossary clarifies the exact meaning of these geometric terms so you never misinterpret an exam question.

Main Content: Key Terms

C–D

Cartesian Equation: An algebraic equation describing a geometric shape using only the spatial coordinates (like $x, y,$ $x, y,$ and $z$ $z$ ), without any parameters like $t$ $t$ or $\lambda$ $λ$ .
- Example: The Cartesian equation of a 2D circle is $x^2 + y^2 = 25$ .
Cauchy–Schwarz Inequality: A fundamental algebraic inequality stating that the absolute value of the dot product of two vectors is always less than or equal to the product of their magnitudes: $|\mathbf{u} \cdot \mathbf{v}| \le |\mathbf{u}| |\mathbf{v}|$ $∣ u \cdot v ∣ \leq ∣ u ∣∣ v ∣$ .
- Example: If $|\mathbf{u}|=3$ and $|\mathbf{v}|=4$ , their dot product can never be greater than $12$ or less than $-12$ .
Collinear: Points that lie on the exact same straight line.
- Example: Points $A(1,1)$ , $B(2,2)$ , and $C(3,3)$ are collinear.
Commutative Law: A mathematical property where changing the order of the operands does not change the result. For vectors, the scalar (dot) product is commutative: $\mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a}$ $a \cdot b = b \cdot a$ .
- Example: $\begin{pmatrix} 1 \\ 2 \end{pmatrix} \cdot \begin{pmatrix} 3 \\ 4 \end{pmatrix} = (1)(3) + (2)(4) = 11$ . Reversing them yields the same $11$ .
Direction Vector: A vector that indicates the orientation or path of a line, but does not define a specific starting point in space. It is usually denoted by $\mathbf{b}$ $b$ in $\mathbf{r} = \mathbf{a} + \lambda\mathbf{b}$ $r = a + λ b$ .
- Example: The line moving from $(1,1)$ to $(4,5)$ has a direction vector of $\begin{pmatrix} 3 \\ 4 \end{pmatrix}$ .
Distributive Law: A property allowing you to expand brackets. For vector dot products: $\mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = \mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c}$ $a \cdot (b + c) = a \cdot b + a \cdot c$ .
- Example: $\mathbf{i} \cdot (\mathbf{i} + \mathbf{j}) = (\mathbf{i} \cdot \mathbf{i}) + (\mathbf{i} \cdot \mathbf{j}) = 1 + 0 = 1$ .

G–P

Gradient: The steepness and direction of a line in 2D space, calculated as "rise over run" ( $m = \frac{\Delta y}{\Delta x}$ $m = \frac{Δ y}{Δ x}$ ). It directly forms a 2D direction vector.
- Example: A line with gradient $m = \frac{3}{2}$ has a direction vector of $\begin{pmatrix} 2 \\ 3 \end{pmatrix}$ .
Gradient–Intercept Form: The standard 2D Cartesian equation of a straight line, written as $y = mx + c$ $y = m x + c$ .
- Example: $y = -2x + 5$ .
Parallel: Lines or vectors that point in exactly the same (or exactly opposite) direction. Mathematically, one vector is a scalar multiple of the other: $\mathbf{u} = k\mathbf{v}$ $u = k v$ .
- Example: The vectors $\begin{pmatrix} 1 \\ 2 \end{pmatrix}$ and $\begin{pmatrix} 3 \\ 6 \end{pmatrix}$ are parallel because the second is $3 \times$ the first.
Parameter: An independent variable (usually $t, \lambda,$ $t, λ,$ or $\mu$ $μ$ ) that is substituted into a vector equation to generate specific points on a line or curve.
- Example: In $\mathbf{r} = \mathbf{a} + \lambda\mathbf{b}$ , if you let the parameter $\lambda = 2$ , you find the point 2 units of direction away from the start.
Position Vector: A vector that defines the exact location of a point in space relative to the origin $O(0,0,0)$ $O (0, 0, 0)$ . Denoted as $\vec{OP}$ $O P$ or simply $\mathbf{p}$ $p$ .
- Example: The point $P(3, -1, 4)$ has the position vector $\mathbf{p} = \begin{pmatrix} 3 \\ -1 \\ 4 \end{pmatrix}$ .

S–V

Scalar: A standard real number with magnitude but no direction.
- Example: $5, -3.2, \text{ and } \pi$ are scalars. $\mathbf{i} + \mathbf{j}$ is not.
Scalar Product: (Also known as the Dot Product). An algebraic operation that takes two vectors and returns a single scalar number. Calculated as $\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3$ $a \cdot b = a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3}$ , or $|\mathbf{a}| |\mathbf{b}| \cos\theta$ $∣ a ∣∣ b ∣ cos θ$ .
- Example: $\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} \cdot \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} = 0$ .
Skew Lines: Two lines in 3-dimensional space that are NOT parallel, but NEVER intersect.
- Example: A line drawn on the floor pointing North, and a line drawn on the ceiling pointing East, are skew.
Vector: A mathematical entity possessing both magnitude (length) and direction.
- Example: Velocity (e.g., $50 \text{ km/h North}$ ).
Vector Equation: An equation defining a line, plane, or curve using vectors and parameters, typically in the form $\mathbf{r} = f(t)$ $r = f (t)$ .
- Example: The vector equation of a circle is $|\mathbf{r} - \mathbf{c}| = R$ .

mini-FAQ page

Q: Are scalar product and dot product exactly the same thing? A: Yes. They are synonymous. It is called the "scalar" product because the result of the calculation is a scalar number, not a vector.

Q: Can skew lines exist in 2D? A: No. In a flat 2D plane, if two lines are not parallel, they MUST eventually intersect. Skew lines are a strictly 3D (or higher) phenomenon.

Common mistakes to avoid

Confusing direction and position vectors: The vector $\begin{pmatrix} 1 \\ 2 \end{pmatrix}$ could be a position (a dot at coordinate $(1,2)$ ) or a direction (an instruction to move 1 right and 2 up). Always check the context of the equation!
Forgetting that parameters must be different for different lines: When trying to find if two lines intersect, if you use $\lambda$ for both $L_1$ and $L_2$ , you are doing it wrong. You must use $\lambda$ for one and $\mu$ for the other.

Practice on Vu's Maths Hub

Language precision is the key to conquering 3D geometry.

Practice converting equations and finding intersections in the HSC Vectors Booklet.
Test your understanding of Cauchy-Schwarz and dot products at Vu's Maths Hub.

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