Welcome to Further Vectors!
In your standard A Level Maths course, you’ve already mastered the basics of vectors in 2D and 3D. In Core Pure Mathematics, we take those concepts and "level up." We aren't just looking at arrows anymore; we are looking at how to describe entire lines and planes (flat surfaces) floating in 3D space.
Understanding these is the secret language behind 3D computer graphics, satellite navigation (GPS), and structural engineering. Don't worry if visualizing 3D space feels a bit brain-bending at first—we'll break it down step-by-step!
1. Straight Lines in 3D Space
To describe a line in 3D, you need two things: a starting point and a direction. Imagine you are at a specific tree (the starting point) and you start walking in a straight line toward a mountain (the direction).
The Vector Equation of a Line
The standard form is:
\(\mathbf{r} = \mathbf{a} + \lambda\mathbf{b}\)
- \(\mathbf{r}\): This represents the position of any point on the line.
- \(\mathbf{a}\): The position vector of a known point on the line (where you start).
- \(\mathbf{b}\): The direction vector (which way the line goes).
- \(\lambda\) (lambda): A scalar parameter. By changing the value of \(\lambda\), you can reach any point on the line.
The Cartesian Equation of a Line
Sometimes, we want the equation in terms of \(x\), \(y\), and \(z\). If \(\mathbf{a} = (a_1, a_2, a_3)\) and \(\mathbf{b} = (b_1, b_2, b_3)\), the equation is:
\(\frac{x-a_1}{b_1} = \frac{y-a_2}{b_2} = \frac{z-a_3}{b_3} = \lambda\)
Memory Aid: Think of this as "rearranging for \(\lambda\)" for each component.
Relationships Between Two Lines
In 2D, lines are either parallel or they meet. In 3D, there is a third, weirder option:
1. Parallel Lines: Their direction vectors (\(\mathbf{b}\)) are multiples of each other.
2. Intersecting Lines: They meet at exactly one point.
3. Skew Lines: They are not parallel, but they never meet. Think of one plane flying at 30,000 feet going North and another at 20,000 feet going East. They pass over/under each other but never crash.
Quick Review: How to find if lines intersect
Step 1: Set the \(x\), \(y\), and \(z\) components of the two lines equal to each other using two different parameters (e.g., \(\lambda\) and \(\mu\)).
Step 2: Solve the first two equations to find \(\lambda\) and \(\mu\).
Step 3: Check if these values work in the third equation. If they do, they intersect. If not, they are skew (unless they were parallel!).
Key Takeaway: A line is just a point plus a multiple of a direction vector. If the directions don't line up and they don't meet, they are skew.
2. The Equation of a Plane
Think of a plane as a flat sheet of paper that goes on forever in all directions. To fix its position in 3D space, we have two main ways to describe it.
The Vector Form
\(\mathbf{r} = \mathbf{a} + \lambda\mathbf{b} + \mu\mathbf{c}\)
To define a flat surface, you need one point (\(\mathbf{a}\)) and two different directions (\(\mathbf{b}\) and \(\mathbf{c}\)) that stay on the surface. By sliding along both directions, you can reach anywhere on the sheet.
The Normal (Scalar Product) Form
This is often the most useful form in Further Maths:
\(\mathbf{r} \cdot \mathbf{n} = k\)
- \(\mathbf{n}\): The normal vector. This is a vector that sticks straight up out of the plane, perfectly perpendicular to it.
- \(k\): A constant calculated by doing \(\mathbf{a} \cdot \mathbf{n}\).
Did you know? A plane's orientation is entirely defined by its normal vector. If you know which way is "straight up" from the surface, you know exactly how the surface is tilted.
The Cartesian Form
If the normal vector \(\mathbf{n} = (a, b, c)\), the equation is:
\(ax + by + cz = d\)
Common Mistake: Students often think \((a, b, c)\) is a point on the plane. It's not! It's the direction of the normal vector. This is a huge help when reading equations—you can see the normal vector immediately!
Key Takeaway: Every plane has a "normal" vector that is perpendicular to it. The coefficients of \(x\), \(y\), and \(z\) in the Cartesian form tell you what that normal vector is.
3. Using the Scalar Product
The scalar product (dot product) is our best tool for checking angles and perpendicularity.
Perpendicularity
Two vectors \(\mathbf{a}\) and \(\mathbf{b}\) are perpendicular (at 90 degrees) if:
\(\mathbf{a} \cdot \mathbf{b} = 0\)
This is vital for checking if a line is perpendicular to a plane or if two planes are perpendicular to each other.
Finding Angles
We use the formula: \(\cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}||\mathbf{b}|}\)
1. Angle between two lines: Use the direction vectors of the lines.
2. Angle between two planes: Use the normal vectors of the planes.
3. Angle between a line and a plane: This is a "trap" for students! Because you are using the line's direction and the plane's normal, the standard formula gives you the angle to the normal. To get the angle to the plane itself, use:
\(\sin \theta = \frac{|\mathbf{d} \cdot \mathbf{n}|}{|\mathbf{d}||\mathbf{n}|}\) (where \(\mathbf{d}\) is the line's direction and \(\mathbf{n}\) is the plane's normal).
Key Takeaway: Use \(\cos\) for Line/Line or Plane/Plane. Use \(\sin\) for Line/Plane.
4. Intersections and Distances
This is where the "Further" in Further Maths really kicks in. We often need to find exactly where things meet or how far apart they are.
Intersection of a Line and a Plane
To find where a line \(\mathbf{r} = \mathbf{a} + \lambda\mathbf{b}\) hits a plane \(ax + by + cz = d\):
Step 1: Write the line's \(x\), \(y\), and \(z\) in terms of \(\lambda\).
Step 2: Substitute these into the plane equation.
Step 3: Solve for \(\lambda\).
Step 4: Plug \(\lambda\) back into the line equation to get the coordinates.
Perpendicular Distance from a Point to a Plane
If you have a point \((\alpha, \beta, \gamma)\) and a plane \(n_1x + n_2y + n_3z + d = 0\), the shortest distance is:
\(dist = \frac{|n_1\alpha + n_2\beta + n_3\gamma + d|}{\sqrt{n_1^2 + n_2^2 + n_3^2}}\)
Analogy: This is like the 2D version you learned in Core Maths, just with an extra \(z\) term! It's essentially "plug the point into the plane and divide by the magnitude of the normal."
Shortest Distance between two Skew Lines
This is the "perpendicular distance" between two lines that don't meet. The shortest path between them will be a line segment that is perpendicular to both.
Note: You will use the scalar product to ensure the connecting vector is perpendicular (\(\cdot = 0\)) to both direction vectors.
Key Takeaway: Most distance problems involve the normal vector because the shortest distance is always a "straight-on" (perpendicular) path.
Summary Checklist
- Can you write a line in \(\mathbf{r} = \mathbf{a} + \lambda\mathbf{b}\) form?
- Do you remember that \(\mathbf{a} \cdot \mathbf{b} = 0\) means 90 degrees?
- Can you pick the normal vector out of \(ax + by + cz = d\)?
- Did you remember to use \(\sin\) for the angle between a line and a plane?
- Don't panic! Drawing a quick sketch of a line hitting a sheet of paper often makes the vector choices much clearer.