Welcome to Further Vectors!

In your earlier maths studies, you’ve likely encountered vectors as simple arrows representing movement. In Further Mathematics (8FM0), we take those arrows and use them to build worlds. We will learn how to describe lines and flat surfaces (planes) in 3D space, find out where they crash into each other, and calculate the "space" or volume they trap. Don't worry if 3D thinking feels a bit like a brain-teaser at first—once you master the formulas, it’s just like following a map!

1. Lines in 3D Space

To describe a line in 3D, you need two things: a starting point (an anchor) and a direction to travel in.

Vector Form

The vector equation of a line is written as:
\(\mathbf{r} = \mathbf{a} + \lambda \mathbf{b}\)

  • \(\mathbf{a}\): The position vector of a point on the line (where you start).
  • \(\mathbf{b}\): The direction vector (which way the line goes).
  • \(\lambda\) (lambda): A scalar parameter. As \(\lambda\) changes, you move along the line.

Cartesian Form

If you prefer coordinates, you can write the line like this:
\(\frac{x - a_1}{b_1} = \frac{y - a_2}{b_2} = \frac{z - a_3}{b_3}\)

Quick Tip: To turn the Cartesian form into the Vector form, just set the whole thing equal to \(\lambda\) and solve for \(x, y,\) and \(z\).

Parallel, Intersecting, or Skew?

In 2D, lines are either parallel or they meet. In 3D, there is a third option:

  • Parallel: The direction vectors are multiples of each other (e.g., one is double the other).
  • Intersecting: They meet at exactly one point.
  • Skew: They aren't parallel, but they never meet. Imagine one plane flying at 30,000 feet going North, and another at 10,000 feet going East. They pass over/under each other but never touch!

Key Takeaway: A 3D line is just a point plus a scaled direction. If direction vectors are not multiples, the lines are either intersecting or skew.

2. The Scalar Product (Dot Product)

The scalar product is a way to multiply two vectors to get a single number (a scalar). It is your best friend for finding angles.

\(\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos \theta\)

Perpendicular Vectors

This is a huge rule for your exams: If two vectors are perpendicular (at 90°), their scalar product is zero.
\(\mathbf{a} \cdot \mathbf{b} = 0 \iff \mathbf{a} \perp \mathbf{b}\)

Did you know? The scalar product tells you how much of one vector is pointing in the same direction as another. If the result is 0, they share nothing in common direction-wise!

3. Equations of a Plane

A plane is a flat, infinite surface (like a sheet of paper that goes on forever).

Vector Form

To define a plane, you need a starting point and two different directions to spread out across the surface:
\(\mathbf{r} = \mathbf{a} + \lambda \mathbf{b} + \mu \mathbf{c}\)

The Normal Form (Scalar Product Form)

A much more common way to write a plane is using a normal vector (\(\mathbf{n}\)). A normal vector is like a flagpole sticking straight up out of the ground at 90° to the surface.
\(\mathbf{r} \cdot \mathbf{n} = d\)

In Cartesian form, this looks like: \(ax + by + cz = d\), where the numbers \((a, b, c)\) are just the components of the normal vector!

Quick Review: To find the equation of a plane, you usually just need a point on the plane and a vector that is 90° to it.

4. The Vector Product (Cross Product)

Note: This is part of the Further Pure 1 (FP1) section of your course.

The vector product (\(\mathbf{a} \times \mathbf{b}\)) is different from the dot product because the answer is a vector, not a number. This new vector is special because it is perpendicular to both original vectors.

Calculating the Cross Product

You can find \(\mathbf{a} \times \mathbf{b}\) using a determinant or the formula from your booklet.
Memory Aid: Think of the "Right-Hand Rule." If your index finger is \(\mathbf{a}\) and your middle finger is \(\mathbf{b}\), your thumb points in the direction of \(\mathbf{a} \times \mathbf{b}\).

Applications: Area

  • Area of a Parallelogram: The magnitude \(|\mathbf{a} \times \mathbf{b}|\) gives you the area of a parallelogram formed by vectors \(\mathbf{a}\) and \(\mathbf{b}\).
  • Area of a Triangle: Since a triangle is half a parallelogram, the area is \(\frac{1}{2} |\mathbf{a} \times \mathbf{b}|\).

Key Takeaway: Use the Cross Product when you need a vector that is 90° to two others, or when you need to find an area.

5. Scalar Triple Product and Volume

When you combine the dot product and the cross product, you get the scalar triple product: \(\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})\).

Applications: Volume

  • Parallelepiped (a 3D squashed box): Volume = \(|\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})|\).
  • Tetrahedron (a 3D triangle/pyramid): Volume = \(\frac{1}{6} |\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})|\).

Common Mistake: Forgetting the \(\frac{1}{6}\) for the tetrahedron! Think of it this way: a tetrahedron is the "pointiest" shape, so it takes up much less space than the full box.

6. Intersections and Distances

Finding the Angle Between a Line and a Plane

When finding the angle between a line (direction \(\mathbf{b}\)) and a plane (normal \(\mathbf{n}\)), use the sine version of the dot product formula:
\(\sin \theta = \frac{|\mathbf{b} \cdot \mathbf{n}|}{|\mathbf{b}| |\mathbf{n}|}\)

Why Sine? Because the dot product gives you the angle with the normal (the flagpole), but we want the angle with the floor. Sine flips the angle correctly for us!

Shortest 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:
\(Distance = \frac{|n_1\alpha + n_2\beta + n_3\gamma + d|}{\sqrt{n_1^2 + n_2^2 + n_3^2}}\)

Step-by-Step for Intersection (Line and Plane):

  1. Write the line in terms of \(\lambda\): \(x = a_1 + \lambda b_1\), \(y = a_2 + \lambda b_2\), etc.
  2. Plug these \(x, y, z\) expressions into the plane's Cartesian equation (\(ax + by + cz = d\)).
  3. Solve for \(\lambda\).
  4. Plug that \(\lambda\) back into the line equation to find the coordinates of the point where they hit!

Summary Checklist

  • Can I find the equation of a line in vector and Cartesian form?
  • Do I remember that \(\mathbf{a} \cdot \mathbf{b} = 0\) means 90°?
  • Can I find the normal vector of a plane from its equation?
  • Do I know the difference between Area (Cross Product) and Volume (Triple Product)?
  • Can I use the point-to-plane distance formula accurately?

Don't worry if this seems tricky at first—vector geometry is all about practice. Keep drawing little sketches of the lines and planes to help your brain visualize the 3D space!