Welcome to Further Vectors!
In your earlier studies, you learned that vectors are essentially "arrows" that tell us about direction and magnitude. In Further Mathematics, we take those arrows and use them to build entire worlds! We will learn how to describe lines and flat surfaces (planes) floating in 3D space. Think of this as the mathematical engine behind GPS, 3D video games, and architectural design.
Don't worry if this seems a bit "mind-bending" at first. Working in 3D is a skill that gets much easier with a little bit of visualization. Let's dive in!
F1: The Equation of a Straight Line in 3D
In 2D, we used \(y = mx + c\). In 3D, we need something more flexible.
The Vector Form
To define a line in space, you only need two things: a starting point and a direction.
The vector equation is: \(\mathbf{r} = \mathbf{a} + \lambda \mathbf{b}\)
- \(\mathbf{r}\): The position vector of any point on the line.
- \(\mathbf{a}\): A known point on the line (the "anchor").
- \(\mathbf{b}\): The direction vector (the way the line is pointing).
- \(\lambda\): A scalar (a number) that tells you how far to move along the direction \(\mathbf{b}\).
The Cartesian Form
If \(\mathbf{a} = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} b_1 \\ b_2 \\ b_3 \end{pmatrix}\), we can rewrite the equation as:
\(\frac{x - a_1}{b_1} = \frac{y - a_2}{b_2} = \frac{z - a_3}{b_3}\)
Quick Review: To turn a vector equation into Cartesian form, just set each component equal to \(\lambda\) and rearrange to make \(\lambda\) the subject!
Common Mistake: Students often mix up the point and the direction. Always ask yourself: "Is this a coordinate on the line, or is it the direction the line is traveling?"
Key Takeaway: A line is just a point plus some "amount" of a direction.
F2: The Equation of a Plane
A plane is a flat, infinite surface. Think of a piece of paper that goes on forever.
The Scalar Product (Normal) Form
The most common way to describe a plane is using a Normal Vector (\(\mathbf{n}\)). A normal vector is a vector that stands 90 degrees (perpendicular) to every single line on the plane.
Analogy: Imagine a flagpole standing straight up on a flat playground. The playground is the plane, and the flagpole is the normal vector.
The equation is: \(\mathbf{r} \cdot \mathbf{n} = d\)
Where \(d = \mathbf{a} \cdot \mathbf{n}\) (using a known point \(\mathbf{a}\) on the plane).
The Cartesian Form of a Plane
If the normal vector \(\mathbf{n} = \begin{pmatrix} a \\ b \\ c \end{pmatrix}\), the equation is simply:
\(ax + by + cz = d\)
Did you know? The coefficients of \(x, y,\) and \(z\) in a plane's equation are actually the components of the vector sticking straight out of it!
Key Takeaway: To find the equation of a plane, your main goal is usually to find its Normal Vector.
F3 & F4: Scalar Products, Angles, and Perpendicularity
The Scalar Product (or Dot Product) is our best tool for finding angles.
\(\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}||\mathbf{b}|\cos\theta\)
Checking for Perpendicular Vectors
If two vectors are perpendicular, the angle between them is \(90^\circ\). Since \(\cos(90^\circ) = 0\):
Two vectors are perpendicular if and only if \(\mathbf{a} \cdot \mathbf{b} = 0\).
Calculating Angles
- Between two lines: Use the dot product of their direction vectors.
- Between two planes: Use the dot product of their normal vectors.
- Between a line and a plane: Use the direction of the line and the normal of the plane. Crucial Step: This gives you the angle with the normal, so you must use \(\sin\theta\) instead of \(\cos\theta\) (or do \(90 - \theta\)).
Key Takeaway: Dot product = 0 means the vectors are perpendicular. It's the "Perpendicularity Test"!
F5: The Vector Product (Cross Product)
The Vector Product \(\mathbf{a} \times \mathbf{b}\) is a bit special. Unlike the scalar product which gives you a number, the vector product gives you a new vector.
Properties of the Vector Product
- The resulting vector is perpendicular to both \(\mathbf{a}\) and \(\mathbf{b}\).
- \(|\mathbf{a} \times \mathbf{b}| = |\mathbf{a}||\mathbf{b}|\sin\theta\).
- If \(\mathbf{a} \times \mathbf{b} = 0\), then the vectors are parallel.
Area of a Triangle
You can use the vector product to find the area of a triangle with sides \(\mathbf{a}\) and \(\mathbf{b}\):
\(Area = \frac{1}{2} |\mathbf{a} \times \mathbf{b}|\)
Alternative Line Equation
A line can also be written as \((\mathbf{r} - \mathbf{a}) \times \mathbf{b} = 0\). This literally says "The vector from our point to any point on the line is parallel to the direction vector."
Memory Aid: Dot product involves Cosine (D-C), Cross product involves Sine (C-S).
F6: Intersections and Distances
This is where we solve "collision" problems.
Intersection of Two Lines
1. Write both lines with different parameters (e.g., \(\lambda\) and \(\mu\)).
2. Set the \(x, y,\) and \(z\) components equal to each other.
3. Solve the resulting equations for \(\lambda\) and \(\mu\).
4. Check: If the values work for all three components, they intersect. If not, the lines are skew (they miss each other in 3D space).
Perpendicular Distance from a Point to a Plane
If you have a point \((x_1, y_1, z_1)\) and a plane \(ax + by + cz = d\), the shortest distance is:
\(Dist = \frac{|ax_1 + by_1 + cz_1 - d|}{\sqrt{a^2 + b^2 + c^2}}\)
Shortest Distance between Two Lines
For skew lines, the shortest distance is along a line that is perpendicular to both. We find this direction using the Vector Product of the two direction vectors.
Encouraging Phrase: Finding the shortest distance between skew lines is one of the toughest parts of this chapter. Take it step-by-step: find a common perpendicular vector first, then project!
Chapter Summary
- Lines: \(\mathbf{r} = \mathbf{a} + \lambda \mathbf{b}\) (Point + Direction).
- Planes: \(\mathbf{r} \cdot \mathbf{n} = d\) (Focus on the Normal vector).
- Scalar Product: Used for angles and perpendicularity (\(\mathbf{a} \cdot \mathbf{b} = 0\)).
- Vector Product: Produces a perpendicular vector; used for areas and parallel checks.
- Intersections: Solve simultaneous equations; remember that 3D lines often don't meet (skew).