Further Vectors

Welcome to Further Vectors!

In your standard A Level course, you’ve already met vectors as a way to describe movement in 2D and 3D. In Further Mathematics, we take those basics and use them to describe entire worlds—specifically, the equations of lines and flat surfaces (planes) in three-dimensional space. Whether you are designing a 3D video game or calculating the flight path of a satellite, these tools are essential. Don’t worry if 3D visualization feels tricky at first; we will break it down into simple, logical steps.

1. Lines in 3D Space

In 2D, you know a line as \(y = mx + c\). In 3D, we need something more flexible. We use two main forms: the Vector Form and the Cartesian Form.

The Vector Equation of a Line

Think of a line as a set of instructions: "Start at this point, and move in this direction."
The equation is: \( \mathbf{r} = \mathbf{a} + \lambda \mathbf{b} \)
• \( \mathbf{a} \) is the position vector of a point on the line (the "starting point").
• \( \mathbf{b} \) is the direction vector (the way the line is pointing).
• \( \lambda \) (lambda) is a scalar parameter. It tells you how many "steps" to take along the direction \( \mathbf{b} \).

The Cartesian Equation of a Line

If you take the vector equation and solve for \( \lambda \) for each component (\(x\), \(y\), and \(z\)), you get the Cartesian form:
\( \frac{x - a_1}{u_1} = \frac{y - a_2}{u_2} = \frac{z - a_3}{u_3} \)
Where \( (a_1, a_2, a_3) \) is your starting point and \( (u_1, u_2, u_3) \) is your direction vector.

Quick Review: To convert from Cartesian to Vector form, just look at the denominators for the direction vector and the numbers being subtracted from \(x, y, z\) for the starting point.

Common Mistake: Always ensure the coefficients of \(x\), \(y\), and \(z\) in the Cartesian form are exactly 1. If you see \( \frac{2x-4}{6} \), you must divide the top and bottom by 2 to get \( \frac{x-2}{3} \) before identifying the direction.

2. The Vector Product (The "Cross Product")

The Scalar Product (Dot Product) gives you a number. The Vector Product (\( \mathbf{a} \times \mathbf{b} \)) gives you a new vector.

Why use it?

The vector product produces a vector that is perpendicular (at 90 degrees) to both original vectors. This is incredibly useful for finding the "normal" to a plane.

How to calculate it:

If \( \mathbf{a} = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix} \) and \( \mathbf{b} = \begin{pmatrix} b_1 \\ b_2 \\ b_3 \end{pmatrix} \), then:
\( \mathbf{a} \times \mathbf{b} = \begin{pmatrix} a_2b_3 - a_3b_2 \\ a_3b_1 - a_1b_3 \\ a_1b_2 - a_2b_1 \end{pmatrix} \)

Did you know? The order matters! \( \mathbf{a} \times \mathbf{b} = -(\mathbf{b} \times \mathbf{a}) \). This is called being anti-commutative. If you swap the order, the resulting vector points in the opposite direction.

3. Equations of a Plane

A plane is a flat, infinite surface. We can describe it in three ways:

A. Vector Form (Two Directions)

\( \mathbf{r} = \mathbf{a} + \lambda \mathbf{b} + \mu \mathbf{c} \)
Here, you start at point \( \mathbf{a} \) and move any distance along two different directions, \( \mathbf{b} \) and \( \mathbf{c} \). Imagine a piece of paper: to cover the whole surface, you can move left/right and up/down.

B. Scalar Product Form (The "Normal" Form)

\( \mathbf{r} \cdot \mathbf{n} = p \)
This is the most common form in Further Maths. \( \mathbf{n} \) is the normal vector (perpendicular to the plane). Any vector \( \mathbf{r} \) on the plane, when dotted with the normal, will always equal the same constant \( p \).

C. Cartesian Form

\( ax + by + cz = d \)
This is the "cleanest" form. The numbers \( a, b, \) and \( c \) are simply the components of the normal vector! So if the normal is \( \begin{pmatrix} 2 \\ -1 \\ 5 \end{pmatrix} \), the plane is \( 2x - y + 5z = d \).

Key Takeaway: The normal vector is your best friend when dealing with planes. If you have two directions on the plane, use the Vector Product to find the normal!

4. Intersections and Angles

Working in 3D involves finding where things meet and how they are tilted relative to each other.

Line-Line Intersections

In 3D, two lines can be:
1. Parallel: Their direction vectors are multiples of each other.
2. Intersecting: They meet at a single point.
3. Skew: This is unique to 3D! They aren't parallel, but they never meet (like two planes flying at different altitudes in different directions).

Calculating Angles

We always use the scalar product formula: \( \cos(\theta) = \frac{|\mathbf{a} \cdot \mathbf{b}|}{|\mathbf{a}||\mathbf{b}|} \).
• Angle between two lines: Use their direction vectors.
• Angle between two planes: Use their normal vectors.
• Angle between a line and a plane: This is the tricky one! Use the line's direction and the plane's normal, but use sin instead of cos: \( \sin(\theta) = \frac{|\mathbf{b} \cdot \mathbf{n}|}{|\mathbf{b}||\mathbf{n}|} \).
Analogy: The normal is at 90 degrees to the plane, so the angle you calculate with cos is the 'wrong' side of the triangle. Using sin corrects this for the angle with the actual plane surface.

5. Shortest Distances

The exam will often ask for the "shortest distance" between points, lines, and planes. These are usually the perpendicular distances. You will be provided with these formulas, but you must know when to use them.

Distance from a Point to a Plane

If the point is \( \mathbf{b} \) and the plane is \( \mathbf{r} \cdot \mathbf{n} = p \):
\( D = \frac{|\mathbf{b} \cdot \mathbf{n} - p|}{|\mathbf{n}|} \)

Distance between Skew Lines

If the lines have points \( \mathbf{a} \) and \( \mathbf{b} \), and a common perpendicular normal \( \mathbf{n} \):
\( D = \frac{|(\mathbf{b} - \mathbf{a}) \cdot \mathbf{n}|}{|\mathbf{n}|} \)
You can find \( \mathbf{n} \) by taking the vector product of the two lines' direction vectors.

Summary Tip: In almost all distance and angle questions, your first step should be to identify the direction vectors for lines and normal vectors for planes.

Quick Review Box:
• Lines: Need a point and a direction.
• Planes: Need a point and a normal.
• Parallel: Directions are multiples.
• Perpendicular: Scalar product = 0.

Don't worry if this seems complex! Practice converting between the different forms of lines and planes first. Once you are comfortable moving between vector and Cartesian forms, the intersection and distance problems become much easier to visualize.