Vectors and three-dimensional coordinate geometry

Welcome to the World of 3D!

In your previous math studies, you’ve likely spent a lot of time in 2D space (the x and y axes). But the real world happens in 3D! In this chapter of the Oxford AQA International Further Mathematics (9665) syllabus, we are adding a third dimension: the z-axis. This is the foundation for everything from architectural design to GPS technology and 3D video game engines. We will learn how to describe lines and flat surfaces (planes) using vectors. Don't worry if it feels like a lot to visualize at first—we'll break it down step-by-step!

1. The Vector Equation of a Line

To define a straight line in 3D, you need two things: a starting point and a direction. Imagine you are telling a friend how to walk in a straight line. You’d say, "Start at the blue bin, and then walk exactly North-East."

In vector form, we write this as:
\( \mathbf{r} = \mathbf{a} + \lambda \mathbf{b} \)

Breaking it down:
• \( \mathbf{r} \): This represents any point \( (x, y, z) \) on the line.
• \( \mathbf{a} \): The position vector of a known point on the line (the "starting point").
• \( \mathbf{b} \): The direction vector (the way the line is pointing).
• \( \lambda \): A scalar parameter. Think of this as a "slider." As you change \( \lambda \), you move back and forth along the line.

Quick Review: Cartesian Form of a Line

You might also see a line written like this:
\( \frac{x - a_1}{b_1} = \frac{y - a_2}{b_2} = \frac{z - a_3}{b_3} \)
Common Mistake: Remember that the numbers on the bottom are the direction components, and the numbers subtracted on top are the coordinates of the point on the line. If you see \( x + 2 \), the coordinate is actually \( -2 \)!

Key Takeaway: A line needs a fixed point and a direction vector. If you have two points, subtract them to find the direction vector!

2. The Scalar Product (Dot Product)

Before moving to planes, we need the Scalar Product. It’s a way of multiplying two vectors to get a single number.

\( \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos \theta \)
Or, if you have coordinates: \( (x_1 x_2) + (y_1 y_2) + (z_1 z_2) \).

Did you know? If the dot product of two vectors is zero, they are perpendicular (at 90 degrees). This is the "Golden Rule" of 3D geometry!

3. The Vector Equation of a Plane

A plane is just a flat, infinite sheet. To pin a plane down in 3D space, we use a Normal Vector (\( \mathbf{n} \)). A normal vector is a vector that sticks straight up out of the plane, like a flagpole on a flat floor.

The Scalar Product Form

The equation is: \( \mathbf{r} \cdot \mathbf{n} = d \)
• \( \mathbf{n} \): The normal vector (perpendicular to the surface).
• \( d \): A constant value calculated by \( \mathbf{a} \cdot \mathbf{n} \) (where \( \mathbf{a} \) is a point on the plane).

The Cartesian Form

This is usually the easiest form to work with: \( ax + by + cz = d \).
In this form, the numbers \( a, b, \) and \( c \) are simply the components of the normal vector! For example, if the normal vector is \( 2\mathbf{i} + 3\mathbf{j} - \mathbf{k} \), the plane equation starts with \( 2x + 3y - z = \dots \)

Analogy: Think of the normal vector as the "DNA" of the plane. It tells us exactly how the plane is tilted in space.

Key Takeaway: To find a plane equation, you need a point on the plane and a normal vector. If you are given two vectors lying *in* the plane, use the cross product to find the normal!

4. Intersections and Angles

This is where we put everything together. Often, the exam will ask how lines and planes interact.

Angle Between Two Lines

Just use the dot product formula on their direction vectors. Ignore the position vectors!

Angle Between a Line and a Plane

Trick Alert! When you find the angle (\( \theta \)) between a line's direction and a plane's normal, you are finding the angle to the "flagpole," not the "floor."
• Use \( \sin \phi = \frac{|\mathbf{b} \cdot \mathbf{n}|}{|\mathbf{b}| |\mathbf{n}|} \) to get the angle with the plane directly.
• Or, find \( \theta \) using cosine and calculate \( 90^\circ - \theta \).

Intersection of a Line and a Plane

1. Write the line in terms of \( \lambda \): \( x = a_1 + \lambda b_1 \), \( y = a_2 + \lambda b_2 \), etc.
2. Substitute these into the plane equation \( ax + by + cz = d \).
3. Solve for \( \lambda \).
4. Plug that \( \lambda \) back into your line equation to find the \( (x, y, z) \) coordinates.

Key Takeaway: Always keep track of whether you are using a direction vector (for lines) or a normal vector (for planes).

5. Perpendicular Distances

Finding the shortest distance from a point to a plane or line is a common exam favorite. Shortest distance always means the perpendicular distance.

Distance from Point \( (x_1, y_1, z_1) \) to Plane \( ax + by + cz = d \)

Use this handy formula:
\( Dist = \frac{|ax_1 + by_1 + cz_1 - d|}{\sqrt{a^2 + b^2 + c^2}} \)

Memory Aid: The denominator is just the magnitude (length) of the normal vector. It’s like the Pythagorean theorem in 3D!

6. Common Pitfalls to Avoid

• Mixing up vectors and points: A point is a location; a vector is a movement. Always check which one you are using.
• Sign errors: When moving from \( \mathbf{r} \cdot \mathbf{n} = d \) to \( ax + by + cz = d \), be careful with negative components in the normal vector.
• The "Sine vs Cosine" confusion: Use cosine for Line-Line or Plane-Plane angles. Use sine for Line-Plane angles.

Final Encouragement: 3D geometry can be hard to visualize. Try using your hands—let your palm be a plane and a pen be a line. Seeing the angles physically can help the math make much more sense!

Chapter Summary Takeaway:
1. Lines: \( \mathbf{r} = \mathbf{a} + \lambda \mathbf{b} \) (Point + Direction).
2. Planes: \( ax + by + cz = d \) (Normal vector coefficients).
3. Perpendicularity: Dot product = 0.
4. Intersections: Substitute line components into the plane equation.