Welcome to Further Vectors!
In your standard A-level Maths course, you’ve likely worked with vectors in 2D or basic 3D. In Further Mathematics, we take those concepts and "level up." We will explore how to describe lines floating in 3D space, how to find the exact point where two flight paths might cross, and how to calculate the shortest distance between objects in space.
Think of this chapter as the mathematics of 3D navigation. Whether it's programming a drone or calculating the path of a satellite, these are the tools engineers use every day!
1. Lines in 3D Space
In 2D, we use \(y = mx + c\). In 3D, that doesn't quite work because a line can tilt in many more directions. Instead, we use Vector and Cartesian forms.
The Vector Equation of a Line
To define a line in 3D, you only need two things:
1. A position vector (\(\mathbf{a}\)): A "starting point" on the line.
2. A direction vector (\(\mathbf{d}\)): The direction the line is traveling in.
The equation is written as:
\(\mathbf{r} = \mathbf{a} + \lambda\mathbf{d}\)
Where \(\mathbf{r}\) represents any point \((x, y, z)\) on the line, and \(\lambda\) (lambda) is a scalar parameter. Think of \(\lambda\) as a "slider"—as you change its value, you move back and forth along the line.
The Cartesian Equation of a Line
If we have a point \((a_1, a_2, a_3)\) and a direction \(\begin{pmatrix} d_1 \\ d_2 \\ d_3 \end{pmatrix}\), we can rewrite the equation by isolating \(\lambda\):
\(\frac{x - a_1}{d_1} = \frac{y - a_2}{d_2} = \frac{z - a_3}{d_3} = \lambda\)
Quick Review Box:
- Vector form: \(\mathbf{r} = (\text{point}) + \lambda(\text{direction})\)
- Cartesian form: All three fractions must be equal to the same constant \(\lambda\).
- Common Mistake: Forgetting that if a direction component is \(0\) (e.g., \(d_2 = 0\)), you can't divide by it. You would write the equation as \(\frac{x - a_1}{d_1} = \frac{z - a_3}{d_3}, y = a_2\).
2. The Scalar Product and Angles
How do we find the angle between two lines in 3D? We use the Scalar Product (also known as the Dot Product).
Calculating the Scalar Product
If you have two vectors \(\mathbf{a} = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} b_1 \\ b_2 \\ b_3 \end{pmatrix}\), the scalar product is:
\(\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3\)
Finding the Angle
The relationship between the dot product and the angle \(\theta\) between two vectors is:
\(\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}||\mathbf{b}| \cos \theta\)
To find the angle between two lines, simply use their direction vectors. It doesn't matter where the lines start; only their directions determine the angle!
Step-by-Step: Finding the Angle
1. Identify the direction vectors \(\mathbf{d_1}\) and \(\mathbf{d_2}\) from the line equations.
2. Calculate the dot product \(\mathbf{d_1} \cdot \mathbf{d_2}\).
3. Calculate the magnitudes (lengths) \(|\mathbf{d_1}|\) and \(|\mathbf{d_2}|\) using Pythagoras: \(\sqrt{x^2 + y^2 + z^2}\).
4. Rearrange the formula: \(\cos \theta = \frac{\mathbf{d_1} \cdot \mathbf{d_2}}{|\mathbf{d_1}||\mathbf{d_2}|}\).
5. Use \(\cos^{-1}\) to find \(\theta\).
Checking for Perpendicular Vectors
This is a very common exam question! If two vectors are perpendicular (at 90 degrees), then \(\cos(90^\circ) = 0\).
Key Takeaway: Two vectors are perpendicular if and only if \(\mathbf{a} \cdot \mathbf{b} = 0\).
3. Intersections and Skew Lines
In 2D, lines are either parallel or they cross. In 3D, there is a third option: Skew lines. These are lines that are not parallel but never meet (like one road passing over another on a bridge).
Finding the Intersection of Two Lines
To find where two lines \(\mathbf{r} = \mathbf{a} + \lambda\mathbf{d}\) and \(\mathbf{r} = \mathbf{b} + \mu\mathbf{e}\) meet:
1. Set the \(x\), \(y\), and \(z\) components of the two lines equal to each other.
2. This gives you three equations with two unknowns (\(\lambda\) and \(\mu\)).
3. Solve two of the equations to find \(\lambda\) and \(\mu\).
4. Crucial Step: Substitute your values into the third equation. If it works, the lines intersect. If it doesn't, the lines are skew.
Analogy: Imagine two airplanes. Even if their paths cross on a 2D map, they only collide if they are at the same altitude (\(z\)-value) at the same time!
4. Perpendicular Distances
Sometimes we need to find the shortest distance from a point to a line, or between two lines. The "shortest distance" is always the perpendicular distance.
Distance from a Point to a Line
To find the distance from point \(P\) to a line with direction \(\mathbf{d}\):
1. Let \(F\) be a general point on the line (expressed in terms of \(\lambda\)).
2. Create a vector for the line segment \(PF\).
3. Since the shortest distance is perpendicular, the dot product of vector \(PF\) and the direction vector \(\mathbf{d}\) must be zero (\(PF \cdot \mathbf{d} = 0\)).
4. Solve for \(\lambda\), find the coordinates of \(F\), and then find the magnitude of vector \(PF\).
Distance Between Two Lines
For two parallel lines, pick any point on one line and find its distance to the other line using the method above.
For skew lines, the process involves finding a vector that is perpendicular to both lines. This is the "common perpendicular."
Memory Aid:
Shortest distance = Perpendicular = Dot Product is Zero. If you are stuck on a distance question, start by setting a dot product to zero!
Summary Checklist
- Can you write a line in \(\mathbf{r} = \mathbf{a} + \lambda\mathbf{d}\) form?
- Do you remember that the angle only depends on the direction vectors?
- Can you prove two vectors are perpendicular by showing their dot product is 0?
- Do you know how to check if lines are skew by testing the third equation?
- Can you find the shortest distance by using the perpendicular property?
Don't worry if the 3D visualization feels tricky at first. Practice sketching the vectors as simple sticks and dots on paper; it helps "see" the math!