Vector equations of lines (HL)

Welcome to the World of Vector Lines!

In your earlier years of mathematics, you likely described lines using the familiar \(y = mx + c\) format. While that works perfectly for 2D graphs, it starts to struggle when we move into three-dimensional space. How do you describe a line floating in the middle of a 3D room? That is where vector equations come to the rescue!

In this chapter, we will learn how to build a line equation that works in both 2D and 3D. Think of a vector equation not as a static "shape," but as a set of instructions for a journey: "Start here, and then head in this direction."

1. The Anatomy of a Vector Equation

The standard form for the vector equation of a line is:

\( \mathbf{r} = \mathbf{a} + \lambda \mathbf{b} \)

Let’s break down what each part means, as this is the foundation for everything else:

\(\mathbf{r}\): This represents the position vector of any generic point on the line. If you pick a value for \(\lambda\), \(\mathbf{r}\) tells you exactly where you are on the line.

\(\mathbf{a}\): This is a known position vector of a specific point on the line. Think of this as your "anchor point" or your starting location.

\(\mathbf{b}\): This is the direction vector. It tells you which way the line is pointing. It’s like the "slope" in \(y = mx + c\), but in vector form.

\(\lambda\) (lambda): This is called the parameter. It is just a scalar (a number). As \(\lambda\) changes, you move back and forth along the line. If \(\lambda = 0\), you are at point \(\mathbf{a}\). If \(\lambda = 1\), you have moved exactly one "length" of vector \(\mathbf{b}\) away from \(\mathbf{a}\).

Quick Review: 2D vs. 3D

The beauty of this formula is that it looks the same regardless of dimensions:

In 2D: \( \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} a_1 \\ a_2 \end{pmatrix} + \lambda \begin{pmatrix} b_1 \\ b_2 \end{pmatrix} \)

In 3D: \( \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix} + \lambda \begin{pmatrix} b_1 \\ b_2 \\ b_3 \end{pmatrix} \)

Key Takeaway: A line is defined by a point it passes through (\(\mathbf{a}\)) and the direction it travels (\(\mathbf{b}\)).

2. How to Find the Equation of a Line

In the IB exams, you are often asked to find the equation of a line passing through two points, say \(A\) and \(B\). Don't worry if this seems tricky; just follow these steps:

Step 1: Pick an anchor point. Use the position vector of either point \(A\) or point \(B\). This will be your \(\mathbf{a}\).

Step 2: Find the direction. To get the direction vector \(\mathbf{b}\), find the displacement vector between the two points: \( \vec{AB} = \mathbf{b} - \mathbf{a} \) (the coordinates of \(B\) minus the coordinates of \(A\)).

Step 3: Assemble. Put them into the formula \( \mathbf{r} = \mathbf{a} + \lambda \mathbf{b} \).

Example: Find the line through \(A(1, 2, 3)\) and \(B(4, 0, -1)\).
1. Anchor point: \( \mathbf{a} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} \)
2. Direction vector: \( \vec{AB} = \begin{pmatrix} 4-1 \\ 0-2 \\ -1-3 \end{pmatrix} = \begin{pmatrix} 3 \\ -2 \\ -4 \end{pmatrix} \)
3. Equation: \( \mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} + \lambda \begin{pmatrix} 3 \\ -2 \\ -4 \end{pmatrix} \)

Did you know? A single line can have many different-looking equations! You could have used point \(B\) as your anchor, or you could have used a direction vector that is twice as long. They all represent the exact same line.

3. The Angle Between Two Lines

When we talk about the angle between two lines, we are really talking about the angle between their direction vectors. We ignore the anchor points (\(\mathbf{a}\)) entirely because the "starting position" doesn't affect how the lines are tilted relative to each other.

If you have two lines:
Line 1: \( \mathbf{r_1} = \mathbf{a_1} + \lambda \mathbf{b_1} \)
Line 2: \( \mathbf{r_2} = \mathbf{a_2} + \mu \mathbf{b_2} \)

Use the scalar product formula you learned in the "Scalar product" chapter:
\( \cos \theta = \frac{|\mathbf{b_1} \cdot \mathbf{b_2}|}{|\mathbf{b_1}| |\mathbf{b_2}|} \)

Important Tip: In Geometry, we usually define the "angle between lines" as the acute angle (the one less than \(90^\circ\)). This is why we use absolute value signs \( | \dots | \) on the top of the fraction—it ensures the result is positive, giving us an acute angle.

4. Simple Applications to Kinematics

In physics and kinematics, vectors are used to describe motion. A vector equation of a line is perfect for an object moving at a constant velocity.

In this context, we usually replace \(\lambda\) with \(t\) (representing time):

\( \mathbf{r} = \mathbf{r_0} + t\mathbf{v} \)

\(\mathbf{r_0}\): The initial position (where the object is at \(t=0\)).
\(\mathbf{v}\): The velocity vector (how fast and in what direction it is moving).
\(t\): The time elapsed.

Example: A particle starts at \( (2, 5) \) and moves with a velocity of \( \begin{pmatrix} 3 \\ -1 \end{pmatrix} \) meters per second. Where is it after 4 seconds?
\( \mathbf{r} = \begin{pmatrix} 2 \\ 5 \end{pmatrix} + 4 \begin{pmatrix} 3 \\ -1 \end{pmatrix} = \begin{pmatrix} 2 + 12 \\ 5 - 4 \end{pmatrix} = \begin{pmatrix} 14 \\ 1 \end{pmatrix} \).
The particle is at the point \( (14, 1) \).

Key Takeaway: In kinematics, the "direction vector" is actually the velocity of the object.

5. Common Mistakes to Avoid

Mistake 1: Confusing Position and Direction.
Remember: \(\mathbf{a}\) is a point on the line. \(\mathbf{b}\) is the direction the line moves. If you swap them, you will get a completely different (and wrong) line!

Mistake 2: Using the wrong vectors for angles.
When finding the angle between lines, always use the direction vectors (\(\mathbf{b}\)). Do not use the position vectors (\(\mathbf{a}\)).

Mistake 3: Forgetting the \(\mathbf{r} =\) part.
An equation must have an equals sign. Don't just write the right-hand side; always start with \( \mathbf{r} = \dots \).

Quick Review Box:
- Vector Equation: \( \mathbf{r} = \mathbf{a} + \lambda \mathbf{b} \)
- To find \(\mathbf{b}\) from two points: \( \mathbf{b} = \text{Point}_2 - \text{Point}_1 \)
- Angle between lines: Use the scalar product on the direction vectors.
- Kinematics: \( \text{Position} = \text{Initial Position} + (\text{Time} \times \text{Velocity}) \).