Welcome to Further Vectors!
In your standard A Level studies, you’ve likely mastered the basics of vectors—adding them, subtracting them, and using the dot product to find angles. Now, we’re stepping into the 3D world of Further Mathematics. We will explore how to find vectors that are perpendicular to others, how to calculate the exact area of shapes floating in 3D space, and how to find the shortest distance between objects that never even touch! Don't worry if it feels like a lot to visualize at first; we will break it down piece by piece.
1. The Vector Product (Cross Product)
In core maths, the Scalar Product (Dot Product) gives you a single number. In Further Maths, we use the Vector Product (also called the Cross Product), which gives you a brand-new vector as the answer.
What does it actually do?
Imagine two vectors, \(\mathbf{a}\) and \(\mathbf{b}\), lying on a flat table. The vector product \(\mathbf{a} \times \mathbf{b}\) is a vector that points straight up (or straight down), perfectly perpendicular to both \(\mathbf{a}\) and \(\mathbf{b}\).
The Geometrical Definition:
\(\mathbf{a} \times \mathbf{b} = |\mathbf{a}||\mathbf{b}| \sin(\theta) \hat{\mathbf{n}}\)
Where \(\hat{\mathbf{n}}\) is a unit vector perpendicular to the plane containing \(\mathbf{a}\) and \(\mathbf{b}\).
How to calculate it (The Formula)
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:
\(\mathbf{a} \times \mathbf{b} = \begin{pmatrix} a_2b_3 - a_3b_2 \\ a_3b_1 - a_1b_3 \\ a_1b_2 - a_2b_1 \end{pmatrix}\)
Memory Aid: The Right-Hand Rule
Point your index finger in the direction of \(\mathbf{a}\) and your middle finger toward \(\mathbf{b}\). Your thumb will now point in the direction of \(\mathbf{a} \times \mathbf{b}\). This is why we call it a right-handed triple!
Important Properties to Remember
- Anti-commutative: \(\mathbf{a} \times \mathbf{b} = -(\mathbf{b} \times \mathbf{a})\). Order matters! If you swap them, the result points in the opposite direction.
- Parallel Vectors: If \(\mathbf{a} \times \mathbf{b} = \mathbf{0}\), then the vectors are parallel. (Analogy: If they point the same way, they don't "spread out" to define a surface, so there's no "up" to point to!)
Quick Review: Use the vector product when you need a vector that is perpendicular to two others.
2. Lines and Planes in 3D
In 3D space, lines and planes are the "building blocks" of geometry. You need to be able to switch between Vector Form and Cartesian Form easily.
The Equation of a Line
Vector Form: \(\mathbf{r} = \mathbf{a} + \lambda \mathbf{u}\)
(\(\mathbf{a}\) is a point on the line; \(\mathbf{u}\) is the direction vector).
Cartesian Form: \(\frac{x - a_1}{u_1} = \frac{y - a_2}{u_2} = \frac{z - a_3}{u_3}\)
The Equation of a Plane
A plane can be defined by a point \(\mathbf{a}\) and a normal vector \(\mathbf{n}\) (a vector sticking straight out of it).
- Vector Form (Standard): \(\mathbf{r} \cdot \mathbf{n} = \mathbf{a} \cdot \mathbf{n}\) (often written as \(\mathbf{r} \cdot \mathbf{n} = d\)).
- Cartesian Form: \(ax + by + cz = d\), where \(\begin{pmatrix} a \\ b \\ c \end{pmatrix}\) is the normal vector.
Did you know? A plane can also be written using two direction vectors (\(\mathbf{b}\) and \(\mathbf{c}\)) that lie on the plane: \(\mathbf{r} = \mathbf{a} + \lambda \mathbf{b} + \mu \mathbf{c}\).
Key Takeaway: The coefficients of \(x, y, z\) in a plane's equation always represent the vector perpendicular to that plane.
3. Intersections and Angles
This is where we test how lines and planes interact in space.
Line-Line Interaction
Two lines in 3D can be:
1. Parallel: Same direction vector.
2. Intersecting: They meet at a single point.
3. Skew: They aren't parallel, but they never meet! (Analogy: Think of one plane flying at 30,000ft going North and another at 20,000ft going East.)
Angles
- Angle between two planes: Find the angle between their normal vectors using the dot product.
- Angle between a line and a plane: Use the dot product between the line’s direction and the plane’s normal. Crucial Trick: The dot product gives you the angle with the normal, so your final answer is \(90^\circ - \theta\).
Common Mistake: Forgetting to do \(90^\circ - \theta\) when finding the angle between a line and a plane. Always sketch it to check!
4. Areas and Volumes (The Scalar Triple Product)
We can use our new vector tools to find the size of 3D shapes.
Areas
- Area of a Parallelogram: \(|\mathbf{a} \times \mathbf{b}|\)
- Area of a Triangle: \(\frac{1}{2} |\mathbf{a} \times \mathbf{b}|\)
Volumes and the Scalar Triple Product
The Scalar Triple Product is written as \(\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})\). It results in a single number (a scalar).
- Volume of a Parallelepiped: \(|\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})|\) (A tilted 3D box).
- Volume of a Tetrahedron: \(\frac{1}{6} |\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})|\) (A triangular pyramid).
Key Concept: Coplanar Vectors
If \(\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = 0\), it means the volume is zero. This only happens if all three vectors lie in the same plane (they are coplanar).
5. Shortest Distances
Finding the gap between objects is a classic exam favorite. You will be given these formulas in the exam, but you must know how to apply them!
1. Distance from a Point to a Plane
\(D = \frac{|\mathbf{b} \cdot \mathbf{n} - p|}{|\mathbf{n}|}\)
Where \(\mathbf{b}\) is the point's position and \(\mathbf{r} \cdot \mathbf{n} = p\) is the plane.
2. Shortest Distance between Skew Lines
\(D = \frac{|(\mathbf{b} - \mathbf{a}) \cdot \mathbf{n}|}{|\mathbf{n}|}\)
Where \(\mathbf{a}\) and \(\mathbf{b}\) are points on each line, and \(\mathbf{n}\) is a vector perpendicular to both lines (found using \(\mathbf{u}_1 \times \mathbf{u}_2\)).
3. Distance from a Point to a Line (2D context)
\(D = \frac{|ax_1 + by_1 - c|}{\sqrt{a^2 + b^2}}\)
Don't worry if these seem tricky! Most distance problems just require you to identify the correct vectors and "plug and play" into the formula. The most important step is finding \(\mathbf{n}\) (the perpendicular vector) using the cross product.
Chapter Summary
- Vector Product (\(\mathbf{a} \times \mathbf{b}\)): Produces a vector perpendicular to both \(\mathbf{a}\) and \(\mathbf{b}\).
- Parallel: Cross product is \(\mathbf{0}\).
- Coplanar: Scalar triple product \(\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})\) is \(0\).
- Area: Relates to the magnitude of the cross product.
- Volume: Relates to the scalar triple product.
- Distances: Use the provided formulas and always find the unit normal vector \(\frac{\mathbf{n}}{|\mathbf{n}|}\) first.