Welcome to the World of 3D Vectors!
In your previous studies, you’ve likely encountered vectors in 2D and basic 3D. In Further Pure Mathematics 3 (FP3), we take these concepts to the next level. We move beyond simple addition and subtraction to explore how vectors can define the very space around us—from the area of a floating triangle to the exact point where two planes in a 3D structure meet.
Whether you’re planning to be an engineer, a video game developer (think 3D graphics!), or a physicist, these tools are your bread and butter. Don't worry if it feels like a lot of steps at first; we will break it down piece by piece!
1. The Vector Product (\(\mathbf{a} \times \mathbf{b}\))
In Core Mathematics, you learned the Scalar Product (Dot Product), which results in a single number. In FP3, we introduce the Vector Product (also called the Cross Product). The big difference? The result is a vector.
What is it?
The vector product \(\mathbf{a} \times \mathbf{b}\) creates a new vector that is perpendicular (at 90 degrees) to both \(\mathbf{a}\) and \(\mathbf{b}\).
How to calculate it
The easiest way to calculate \(\mathbf{a} \times \mathbf{b}\) for vectors \(\mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} + a_3\mathbf{k}\) and \(\mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j} + b_3\mathbf{k}\) is to use a determinant:
\[ \mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix} \]
Step-by-step:
1. For the \(\mathbf{i}\) component: \((a_2b_3 - a_3b_2)\)
2. For the \(\mathbf{j}\) component: -(\(a_1b_3 - a_3b_1\)) (Notice the minus sign! This is a common mistake.)
3. For the \(\mathbf{k}\) component: \((a_1b_2 - a_2b_1)\)
Geometric Interpretation: Area
Did you know? The magnitude of the cross product, \(|\mathbf{a} \times \mathbf{b}|\), is equal to the area of the parallelogram formed by the two vectors.
If you only want the area of the triangle formed by them, just divide by 2!
Area of Triangle = \(\frac{1}{2} |\mathbf{a} \times \mathbf{b}|\)
Quick Review:
- \(\mathbf{a} \times \mathbf{b}\) is a vector.
- It is perpendicular to both original vectors.
- Magnitude = Area of the parallelogram.
2. The Triple Scalar Product (\(\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})\))
This sounds fancy, but it's just a dot product combined with a cross product. The result is a scalar (a number).
Geometric Interpretation: Volume
Imagine a squashed 3D box where all sides are parallelograms. This is called a parallelepiped. The volume of this shape is the absolute value of the triple scalar product: \(|\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})|\).
If the result is zero, it means the vectors \(\mathbf{a}\), \(\mathbf{b}\), and \(\mathbf{c}\) all lie in the same flat plane. We call these coplanar vectors.
Memory Aid:
- 1 Vector = A line
- 2 Vectors (Cross product) = Area (2D)
- 3 Vectors (Triple product) = Volume (3D)
3. Equations of Lines
You already know the standard vector equation of a line: \(\mathbf{r} = \mathbf{a} + \lambda \mathbf{b}\). In FP3, we see a new way to write this using the cross product.
The Cross Product Form
A line can be written as: \((\mathbf{r} - \mathbf{a}) \times \mathbf{b} = \mathbf{0}\)
Why does this work?
Think about it: \((\mathbf{r} - \mathbf{a})\) is a vector pointing from a fixed point \(\mathbf{a}\) to any point \(\mathbf{r}\) on the line. If this vector is parallel to the direction vector \(\mathbf{b}\), their cross product must be zero. It's just a different way of saying the points are in a straight line!
4. Equations of Planes
A plane is a flat, infinite 2D surface in 3D space. There are three main ways to describe one:
A. Parametric Form
\(\mathbf{r} = \mathbf{a} + s\mathbf{b} + t\mathbf{c}\)
Here, \(\mathbf{a}\) is a point on the plane, and \(\mathbf{b}\) and \(\mathbf{c}\) are two vectors that "lie" in the plane. By changing the "sliders" \(s\) and \(t\), you can reach any point on the surface.
B. Scalar Product (Normal) Form
\(\mathbf{r} \cdot \mathbf{n} = p\)
This is the most common form. \(\mathbf{n}\) is the normal vector (a vector sticking straight up out of the plane, like a flagpole on the ground). \(p\) is a constant.
C. Cartesian Form
\(ax + by + cz = d\)
In this form, the coefficients \((a, b, c)\) are actually the components of the normal vector \(\mathbf{n}\). So if you see the plane \(2x - 3y + z = 5\), you immediately know the normal vector is \(2\mathbf{i} - 3\mathbf{j} + \mathbf{k}\)!
5. Solving Geometric Problems
This is where FP3 gets exciting! We use these formulas to find distances and intersections.
I. Distance from a Point to a Plane
If you have a point \(P\) with position vector \(\mathbf{x_0}\) and a plane \(\mathbf{r} \cdot \mathbf{n} = p\), the shortest distance is:
\[ d = \frac{|\mathbf{x_0} \cdot \mathbf{n} - p|}{|\mathbf{n}|} \]
II. Line of Intersection of Two Planes
When two planes meet (like two walls in a room), they form a line. To find the direction of this line, you find the cross product of the two normal vectors: \(\mathbf{d} = \mathbf{n_1} \times \mathbf{n_2}\).
III. Shortest Distance Between Two Skew Lines
Skew lines are lines that are not parallel but never meet because they are at different "depths" in 3D space. To find the shortest gap between them:
1. Find a vector perpendicular to both lines using \(\mathbf{n} = \mathbf{b_1} \times \mathbf{b_2}\) (where \(\mathbf{b}\) are the direction vectors).
2. The distance is the projection of the vector connecting any two points on the lines onto this normal vector.
Common Mistake:
Don't confuse parallel lines with skew lines. Parallel lines have the same direction vector. Skew lines have different direction vectors but still don't intersect.
Summary and Key Takeaways
Key Points to Remember:
- Cross Product: Result is a vector. Use it for perpendiculars and areas.
- Triple Product: Result is a scalar. Use it for volumes and checking if vectors are coplanar.
- Normal Vector: The "key" to any plane. It is always perpendicular to the surface.
- Equation of a Line: Can be written as \((\mathbf{r} - \mathbf{a}) \times \mathbf{b} = \mathbf{0}\).
- Equation of a Plane: \(\mathbf{r} \cdot \mathbf{n} = p\) is your best friend for distance problems.
Don't worry if this seems tricky at first! Vectors in 3D require a bit of spatial imagination. Try drawing a quick sketch of the planes and lines when you start a problem—it helps more than you might think!