Welcome to Further Vectors!
In your previous math studies, you likely met vectors as arrows showing direction and magnitude. In Further Mathematics (9231), we take those arrows and use them to build 3D worlds! We will learn how to describe flat surfaces (called planes) and figure out exactly how lines and planes interact in space.
Don’t worry if 3D visualization feels tough at first—most students find it tricky! We will use clear steps and some handy tricks to make these spatial puzzles much easier to solve.
1. The Vector Product (The Cross Product)
Previously, you learned the Scalar Product (Dot Product), which gives you a number. Now, meet the Vector Product (Cross Product), which gives you a brand new vector!
What is it?
The vector product of two vectors \(\mathbf{a}\) and \(\mathbf{b}\) is written as \(\mathbf{a} \times \mathbf{b}\). This new vector has a very special property: it is perpendicular (at 90°) to both \(\mathbf{a}\) and \(\mathbf{b}\).
How to Calculate It
There are two ways to look at it:
1. The Geometric Way: \(\mathbf{a} \times \mathbf{b} = (|\mathbf{a}||\mathbf{b}| \sin \theta) \mathbf{\hat{n}}\)
Where \(\theta\) is the angle between them and \(\mathbf{\hat{n}}\) is a unit vector pointing in the perpendicular direction.
2. The Component Way (The most useful for exams!):
If \(\mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} + a_3\mathbf{k}\) and \(\mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j} + b_3\mathbf{k}\), then:
\(\mathbf{a} \times \mathbf{b} = (a_2b_3 - a_3b_2)\mathbf{i} + (a_3b_1 - a_1b_3)\mathbf{j} + (a_1b_2 - a_2b_1)\mathbf{k}\)
Quick Trick: Think of this as a 3x3 determinant. If you find the formula hard to memorize, write the components of \(\mathbf{a}\) and \(\mathbf{b}\) in two rows and use the "cover-up" method you use for matrices!
Did you know? You can find the direction of \(\mathbf{a} \times \mathbf{b}\) using the Right-Hand Rule. Point your fingers from \(\mathbf{a}\) towards \(\mathbf{b}\), and your thumb points in the direction of the cross product!
Key Takeaway:
The Vector Product creates a vector that is perpendicular to the plane containing the original two vectors. If \(\mathbf{a} \times \mathbf{b} = 0\), the vectors are parallel.
2. The Equations of a Plane
A plane is a flat, infinite surface. In this course, you need to be comfortable switching between three different ways of writing a plane's equation.
Form 1: The Vector/Parametric Form
\(\mathbf{r} = \mathbf{a} + \lambda\mathbf{b} + \mu\mathbf{c}\)
Analogy: Imagine standing at a point (vector \(\mathbf{a}\)) on a giant floor. To reach any other spot on that floor, you can walk some distance in one direction (\(\lambda\mathbf{b}\)) and some distance in another direction (\(\mu\mathbf{c}\)).
Form 2: The Scalar Product Form
\(\mathbf{r} \cdot \mathbf{n} = d\)
Here, \(\mathbf{n}\) is the normal vector (a vector sticking straight up out of the plane at 90°). This is the most powerful form for solving problems!
Form 3: The Cartesian Form
\(ax + by + cz = d\)
This is just the scalar product multiplied out! The numbers \(a\), \(b\), and \(c\) are simply the components of the normal vector \(\mathbf{n} = \begin{pmatrix} a \\ b \\ c \end{pmatrix}\).
Common Mistake: Students often think the vector in the Cartesian equation is inside the plane. Nope! The coefficients \(a, b, c\) always represent the Normal (the direction pointing away from the plane).
Quick Review: How to convert?
To go from Parametric (\(\mathbf{b}\) and \(\mathbf{c}\)) to Cartesian, simply find the cross product \(\mathbf{b} \times \mathbf{c}\). This gives you your normal vector \(\mathbf{n}\)!
3. Intersections and Relationships
Now we play "spatial detective." How do lines and planes interact?
Line and Plane Intersection
To find where a line \(\mathbf{r} = \mathbf{a} + \lambda\mathbf{d}\) hits a plane \(ax + by + cz = d\):
1. Write the \(x, y, z\) components of the line in terms of \(\lambda\).
2. Plug these into the plane equation.
3. Solve for \(\lambda\).
4. Put \(\lambda\) back into the line equation to get the point of intersection.
Parallel or Lying In?
Check the dot product of the line's direction \(\mathbf{d}\) and the plane's normal \(\mathbf{n}\):
- If \(\mathbf{d} \cdot \mathbf{n} = 0\), the line is perpendicular to the normal. This means the line is either parallel to the plane or lying inside it.
- To check which one, see if the starting point of the line satisfies the plane's equation!
4. Angles in 3D Space
When calculating angles, always identify which "directions" you are using.
Angle between two Planes
The angle between two planes is the same as the angle between their normal vectors (\(\mathbf{n_1}\) and \(\mathbf{n_2}\)). Use the dot product formula:
\(\cos \theta = \frac{|\mathbf{n_1} \cdot \mathbf{n_2}|}{|\mathbf{n_1}||\mathbf{n_2}|}\)
Angle between a Line and a Plane
Watch out! This is a classic exam trap. The dot product between the line (\(\mathbf{d}\)) and the normal (\(\mathbf{n}\)) gives you the angle with the normal, not the floor. To get the angle with the plane, we use Sine:
\(\sin \theta = \frac{|\mathbf{d} \cdot \mathbf{n}|}{|\mathbf{d}||\mathbf{n}|}\)
Key Takeaway:
Plane-Plane uses Cos. Line-Plane uses Sin. Always use absolute values for the dot product to ensure you get the acute angle!
5. Distances and Perpendiculars
Foot of the Perpendicular from a Point to a Plane
Imagine dropping a ball from a point \(P\) onto a floor (plane). The "Foot" \(F\) is where it hits.
Step-by-Step:
1. Create a line that passes through \(P\) and has the same direction as the plane's normal \(\mathbf{n}\).
2. Find where this line intersects the plane (using the intersection method from Section 3).
3. The resulting point is the Foot of the Perpendicular.
Shortest Distance between Two Skew Lines
Skew lines are lines that are not parallel but never meet (like two planes flying at different heights in different directions).
The shortest distance is measured along a line that is perpendicular to both.
1. Find the common perpendicular direction: \(\mathbf{n} = \mathbf{d_1} \times \mathbf{d_2}\).
2. The distance is the projection of the vector connecting any two points on the lines onto this normal.
Formula: \(Dist = \frac{|(\mathbf{a_1} - \mathbf{a_2}) \cdot \mathbf{n}|}{|\mathbf{n}|}\)
6. Summary Checklist
Before your exam, make sure you can:
- Calculate \(\mathbf{a} \times \mathbf{b}\) quickly and accurately.
- Find the normal vector of a plane from its equation.
- Convert \(\mathbf{r} = \mathbf{a} + \lambda\mathbf{b} + \mu\mathbf{c}\) into \(ax + by + cz = d\).
- Use Sin for line-plane angles and Cos for plane-plane angles.
- Find the line of intersection of two planes (Hint: Set \(z=0\) to find one point, then cross the normals to find the direction!).
Don't worry if this seems like a lot of steps! Vectors in Further Maths is all about practice. Once you realize that the Normal Vector is the "key" to almost every problem, everything starts to click!