👋 Welcome to the World of 3D Vectors!
Hello future P4 mathematician! This chapter, Vectors, is one of the most powerful and exciting topics you'll study. Vectors allow us to describe movement, forces, and position in three dimensions (3D). If you plan on studying engineering, physics, or computer graphics, this is essential knowledge!
Don't worry if coordinates and directions seem complicated in 3D space at first. We will break down every concept step-by-step, using simple analogies to help you master lines and relationships in space. Let’s get started!
📐 Section 1: Fundamentals of Vectors in 3D Space
What is a Vector? (A Quick Review)
In simple terms, a vector is a quantity that has both magnitude (size/length) and direction.
- Scalar: Only magnitude (e.g., speed, temperature, time, mass).
- Vector: Magnitude AND direction (e.g., velocity, force, displacement).
In P4, we primarily deal with vectors in 3D space, which means we now introduce the z-axis.
Notation and Components
A vector \(\mathbf{a}\) in 3D space is defined by its components along the x, y, and z axes. We use the standard basis vectors:
- \(\mathbf{i}\): Unit vector in the direction of the positive x-axis.
- \(\mathbf{j}\): Unit vector in the direction of the positive y-axis.
- \(\mathbf{k}\): Unit vector in the direction of the positive z-axis.
A general vector \(\mathbf{a}\) can be written in two ways:
1. Component Form:
$$\mathbf{a} = x \mathbf{i} + y \mathbf{j} + z \mathbf{k}$$
2. Column Vector Form (preferred for calculations):
$$\mathbf{a} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}$$
Analogy: Think of \(\mathbf{i}\) as "forward/back", \(\mathbf{j}\) as "left/right", and \(\mathbf{k}\) as "up/down" in a large warehouse. To get to a specific point, you need all three instructions!
Position and Displacement Vectors
Position Vector (\(\mathbf{r}\))
A position vector describes the location of a point P relative to a fixed origin O. It is usually denoted \(\mathbf{r}\) or \(\mathbf{OP}\).
If point \(P\) has coordinates \((x, y, z)\), then its position vector is: $$\mathbf{p} = \vec{OP} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}$$
Displacement Vector (Finding the Vector Between Two Points)
The displacement vector describes the journey from one point (A) to another (B).
If \(\mathbf{a}\) is the position vector of A, and \(\mathbf{b}\) is the position vector of B, the vector from A to B is found using the rule:
The Head Minus Tail Rule: $$\vec{AB} = \mathbf{b} - \mathbf{a}$$
Quick Tip: To remember this, always subtract the vector of the starting point (Tail) from the vector of the end point (Head).
Magnitude (Length) of a Vector
The magnitude of a vector is its length, often denoted by the notation \(|\mathbf{a}|\). Since the components are perpendicular, we use the 3D version of the Pythagorean theorem.
If \(\mathbf{a} = x \mathbf{i} + y \mathbf{j} + z \mathbf{k}\), then: $$|\mathbf{a}| = \sqrt{x^2 + y^2 + z^2}$$
Memory Aid: Magnitude is always positive! You are finding a distance.
Unit Vectors
A unit vector is a vector that has a magnitude of exactly 1.
To find the unit vector in the direction of \(\mathbf{a}\), you simply divide the vector \(\mathbf{a}\) by its own magnitude:
$$\hat{\mathbf{a}} = \frac{\mathbf{a}}{|\mathbf{a}|}$$
(The little hat, \(\hat{\mathbf{a}}\), is often used to denote a unit vector.)
🔑 Key Takeaway: Section 1
We work in 3D (\(x, y, z\)). Displacement is End minus Start. Magnitude uses Pythagoras in 3D. A unit vector has a length of 1.
➕ Section 2: Vector Arithmetic (The Basics)
Vector Addition and Subtraction
Adding and subtracting vectors is straightforward: you simply add or subtract the corresponding components.
If \(\mathbf{a} = \begin{pmatrix} x_1 \\ y_1 \\ z_1 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} x_2 \\ y_2 \\ z_2 \end{pmatrix}\):
Addition: $$\mathbf{a} + \mathbf{b} = \begin{pmatrix} x_1 + x_2 \\ y_1 + y_2 \\ z_1 + z_2 \end{pmatrix}$$ Subtraction: $$\mathbf{a} - \mathbf{b} = \begin{pmatrix} x_1 - x_2 \\ y_1 - y_2 \\ z_1 - z_2 \end{pmatrix}$$
Did you know? Geometrically, vector addition follows the "Triangle Rule" or the "Parallelogram Rule." Adding \(\mathbf{a} + \mathbf{b}\) means following the path \(\mathbf{a}\) and then following the path \(\mathbf{b}\).
Scalar Multiplication (Scaling a Vector)
When you multiply a vector \(\mathbf{a}\) by a scalar (a simple number, usually denoted \(\lambda\) or \(k\)), you multiply every single component of the vector by that scalar.
$$k \mathbf{a} = k \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} kx \\ ky \\ kz \end{pmatrix}$$
The resulting vector will be parallel to \(\mathbf{a}\).
Key Property: Parallel Vectors
Two non-zero vectors \(\mathbf{a}\) and \(\mathbf{b}\) are parallel if and only if one is a scalar multiple of the other, i.e., \(\mathbf{a} = k \mathbf{b}\).
⚠️ Common Mistake Alert!
When calculating a scalar multiple like \(3(\mathbf{a} - \mathbf{b})\), remember to subtract \(\mathbf{b}\) from \(\mathbf{a}\) first, then multiply the resulting vector by 3.
• Section 3: The Scalar Product (Dot Product)
The scalar product (or dot product) is an operation between two vectors that yields a scalar (a number). It is denoted by a large dot: \(\mathbf{a} \cdot \mathbf{b}\).
Calculating the Scalar Product in Component Form
To find the scalar product of two vectors, you multiply their corresponding components and add the results.
If \(\mathbf{a} = \begin{pmatrix} x_1 \\ y_1 \\ z_1 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} x_2 \\ y_2 \\ z_2 \end{pmatrix}\):
$$\mathbf{a} \cdot \mathbf{b} = x_1 x_2 + y_1 y_2 + z_1 z_2$$
Geometric Definition and Angle Calculation
The dot product is hugely important because it relates the vectors' lengths to the angle between them.
The geometric definition is: $$\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos \theta$$ where \(\theta\) is the angle between the two vectors (\(0^\circ \leq \theta \leq 180^\circ\)).
We can rearrange this formula to find the angle \(\theta\): $$\cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| |\mathbf{b}|}$$
Step-by-Step: Finding the Angle (\(\theta\))
- Calculate the dot product (\(\mathbf{a} \cdot \mathbf{b}\)).
- Calculate the magnitude of \(\mathbf{a}\) and \(\mathbf{b}\) (\(|\mathbf{a}|\) and \(|\mathbf{b}|\)).
- Substitute into the \(\cos \theta\) formula.
- Use \(\theta = \cos^{-1}(\dots)\) to find the angle.
The Test for Perpendicularity (Orthogonality)
This is the most critical application of the dot product in P4.
If two vectors \(\mathbf{a}\) and \(\mathbf{b}\) are perpendicular (meet at \(90^\circ\)), then \(\cos 90^\circ = 0\).
Therefore, for perpendicular vectors: $$\mathbf{a} \cdot \mathbf{b} = 0$$
Memory Aid: When vectors are perpendicular, they are "DOTally zero." If you need to prove or find an unknown component that makes two vectors perpendicular, set their dot product to zero.
🔑 Key Takeaway: Section 3
The scalar product (\(\mathbf{a} \cdot \mathbf{b}\)) gives us a number. Its main use is finding the angle between vectors, especially checking if that angle is \(90^\circ\). If \(\mathbf{a} \cdot \mathbf{b} = 0\), the vectors are perpendicular.
📍 Section 4: Lines in 3D Space
Unlike 2D lines, 3D lines are usually expressed using vectors. We need a fixed point on the line and a direction to describe it fully.
The Vector Equation of a Line
The position vector \(\mathbf{r}\) of any point on a line is given by:
$$\mathbf{r} = \mathbf{a} + \lambda \mathbf{b}$$
Where:
- \(\mathbf{r}\): The running position vector (describes any point on the line).
- \(\mathbf{a}\): A specific position vector of a known point on the line (the starting point).
- \(\mathbf{b}\): The direction vector (tells the line where to go).
- \(\lambda\): The scalar parameter (a variable number that scales the direction, allowing us to reach all points on the line).
Analogy: \(\mathbf{a}\) is your address (fixed location). \(\mathbf{b}\) is your GPS instruction ("go 3 blocks east, 1 block up"). \(\lambda\) is how long you follow the instruction (travel time).
The Cartesian Equation of a Line
While the vector form is cleaner, the Cartesian form is essential for finding intersection points.
Starting with the vector form: $$\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix} + \lambda \begin{pmatrix} b_1 \\ b_2 \\ b_3 \end{pmatrix}$$
We separate this into three parametric equations: $$x = a_1 + \lambda b_1 \quad \rightarrow \quad \lambda = \frac{x - a_1}{b_1}$$ $$y = a_2 + \lambda b_2 \quad \rightarrow \quad \lambda = \frac{y - a_2}{b_2}$$ $$z = a_3 + \lambda z_3 \quad \rightarrow \quad \lambda = \frac{z - a_3}{b_3}$$
Since \(\lambda\) must be the same for a single point, we equate them all to get the Cartesian form:
$$\frac{x - a_1}{b_1} = \frac{y - a_2}{b_2} = \frac{z - a_3}{b_3}$$
Case of a Zero Direction Component
If a direction vector component is zero (e.g., \(b_1 = 0\)), the line is parallel to the corresponding plane. The equation for that variable simply states the fixed coordinate:
If \(\mathbf{b} = \begin{pmatrix} 0 \\ 4 \\ 7 \end{pmatrix}\) and \(\mathbf{a} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}\), the line is: $$\mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} + \lambda \begin{pmatrix} 0 \\ 4 \\ 7 \end{pmatrix}$$ The Cartesian form becomes: $$x = 1; \quad \frac{y - 2}{4} = \frac{z - 3}{7}$$
Finding the Angle Between Two Lines
The angle between two lines, \(L_1\) and \(L_2\), is defined as the angle between their direction vectors.
If \(L_1\) has direction vector \(\mathbf{b}_1\) and \(L_2\) has direction vector \(\mathbf{b}_2\), we use the dot product formula:
$$\cos \theta = \frac{|\mathbf{b}_1 \cdot \mathbf{b}_2|}{|\mathbf{b}_1| |\mathbf{b}_2|}$$
Crucial Note on Acute Angle: When finding the angle between lines, we usually specify the acute angle (\(0^\circ \leq \theta \leq 90^\circ\)). To guarantee an acute angle, we take the absolute value of the dot product in the numerator (that's why the formula has \(|\dots|\)).
Intersection of Two Lines
To find if two lines, \(L_1\) and \(L_2\), intersect, we must find a set of parameters (\(\lambda\) for \(L_1\), and a different parameter, say \(\mu\), for \(L_2\)) that make the position vectors equal.
Given: $$L_1: \mathbf{r}_1 = \mathbf{a}_1 + \lambda \mathbf{b}_1$$ $$L_2: \mathbf{r}_2 = \mathbf{a}_2 + \mu \mathbf{b}_2$$
Step-by-Step Intersection Check:
- Equate the vectors: \(\mathbf{a}_1 + \lambda \mathbf{b}_1 = \mathbf{a}_2 + \mu \mathbf{b}_2\).
- Form three simultaneous equations: One for \(x\), one for \(y\), and one for \(z\).
- Solve two equations simultaneously (e.g., using the \(x\) and \(y\) equations) to find unique values for \(\lambda\) and \(\mu\).
- Check Consistency: Substitute these values of \(\lambda\) and \(\mu\) into the third equation (the \(z\) equation).
- If the third equation holds true (LHS = RHS), the lines intersect. If it does not hold true, the lines do not intersect (they are skew lines).
- If they intersect, substitute the found \(\lambda\) (or \(\mu\)) back into its original line equation to find the coordinates of the intersection point.
🔑 Key Takeaway: Section 4
A line requires a starting point (\(\mathbf{a}\)) and a direction (\(\mathbf{b}\)). The angle between lines depends only on their direction vectors. Intersection requires checking for consistency across all three dimensions using two different parameters (\(\lambda\) and \(\mu\)).
🌟 Final Words of Encouragement
Vectors is a chapter where visualization is key! If you struggle, try sketching a simple coordinate system (even just the positive axes) and tracing the paths. Focus on the core relationships: magnitude (Pythagoras), angle (Dot Product), and line definition (\(\mathbf{r} = \mathbf{a} + \lambda \mathbf{b}\)). You’ve got this!