Welcome to Further Vectors!
In your earlier math journey, you’ve likely worked with vectors in 2D. Now, we are stepping into the 3D world! This chapter is part of your Additional Pure Mathematics section. We will learn how to describe lines in space, find out if they crash into each other, and explore two different ways to "multiply" vectors to find angles and areas.
Don’t worry if this seems a bit abstract at first. Think of 3D vectors as instructions for a drone: go forward, go right, and go up. Once you get the hang of the notation, it’s just like a puzzle!
1. Vector Equations of Straight Lines
To define a line in 3D space, you need two things: a starting point and a direction.
The Vector Form
The equation of a line is written as:
\(\mathbf{r} = \mathbf{a} + \lambda\mathbf{b}\)
- \(\mathbf{r}\): The position of any point on the line.
- \(\mathbf{a}\): A fixed point that the line passes through (the "anchor").
- \(\mathbf{b}\): The direction vector (where the line is heading).
- \(\lambda\): A scalar parameter (think of this as a "slider" that moves you along the line).
The Cartesian Form
Sometimes we write the line using \(x, y,\) and \(z\). For a line passing through \((a_1, a_2, a_3)\) with direction vector \(\begin{pmatrix} u_1 \\ u_2 \\ u_3 \end{pmatrix}\), the form is:
\(\frac{x - a_1}{u_1} = \frac{y - a_2}{u_2} = \frac{z - a_3}{u_3} = \lambda\)
Quick Tip: To switch between forms, just set each part of the Cartesian form equal to \(\lambda\) and solve for \(x, y,\) and \(z\).
Key Takeaway: A line is just a point plus "some amount" of a direction. If you have two points, your direction vector \(\mathbf{b}\) is simply the difference between them!
2. The Scalar Product (Dot Product)
The scalar product (written as \(\mathbf{a} \cdot \mathbf{b}\)) is a way to multiply two vectors to get a single number (a scalar).
How to Calculate it
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:
\(\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3\)
Why is it useful?
We use it to find the angle between two vectors using this formula:
\(\cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}||\mathbf{b}|}\)
Important Property: Perpendicularity
If two vectors are perpendicular (at 90 degrees), their scalar product is zero.
\(\mathbf{a} \cdot \mathbf{b} = 0 \iff \mathbf{a} \perp \mathbf{b}\)
Common Mistake: Students often forget that the scalar product gives a number, not a vector. If your answer has an \(\mathbf{i, j,}\) or \(\mathbf{k}\) in it, you've done something wrong!
Key Takeaway: Use the Dot Product when you need to find an angle or check if lines are perpendicular.
3. Intersections and Skew Lines
In 2D, lines are either parallel or they meet. In 3D, there is a third option: Skew lines.
- Parallel lines: Their direction vectors are multiples of each other.
- Intersecting lines: They meet at exactly one point.
- Skew lines: They are not parallel, but they never meet (like two planes flying at different altitudes in different directions).
How to find the intersection:
- Set the \(x, y,\) and \(z\) components of two line equations equal to each other (using different parameters, like \(\lambda\) and \(\mu\)).
- Solve the first two equations to find \(\lambda\) and \(\mu\).
- The Check: Plug your values into the third (\(z\)) equation. If it works, they intersect! If it doesn't, they are skew.
Key Takeaway: Always check the third equation! If \(1=5\), don't panic—it just means the lines are skew.
4. The Vector Product (Cross Product)
Unlike the scalar product, the vector product (written \(\mathbf{a} \times \mathbf{b}\)) results in a new vector.
What does it represent?
The result of \(\mathbf{a} \times \mathbf{b}\) is a vector that is perpendicular to both \(\mathbf{a}\) and \(\mathbf{b}\).
Did you know? This follows the Right-Hand Rule. If you curl the fingers of your right hand from \(\mathbf{a}\) to \(\mathbf{b}\), your thumb points in the direction of \(\mathbf{a} \times \mathbf{b}\).
The Formula
The geometric definition is:
\(\mathbf{a} \times \mathbf{b} = |\mathbf{a}||\mathbf{b}| \sin \theta \hat{\mathbf{n}}\)
where \(\hat{\mathbf{n}}\) is the unit vector perpendicular to the plane containing \(\mathbf{a}\) and \(\mathbf{b}\).
How to calculate it (The Determinant Method)
To calculate \(\mathbf{a} \times \mathbf{b}\), use the formula provided in your booklet:
\(\mathbf{a} \times \mathbf{b} = \begin{pmatrix} a_2b_3 - a_3b_2 \\ a_3b_1 - a_1b_3 \\ a_1b_2 - a_2b_1 \end{pmatrix}\)
Properties to Remember:
- Anti-commutative: \(\mathbf{a} \times \mathbf{b} = -(\mathbf{b} \times \mathbf{a})\). Order matters!
- Parallel vectors: If \(\mathbf{a}\) and \(\mathbf{b}\) are parallel, \(\mathbf{a} \times \mathbf{b} = \mathbf{0}\) (the zero vector).
- Distributive: \(\mathbf{a} \times (\mathbf{b} + \mathbf{c}) = \mathbf{a} \times \mathbf{b} + \mathbf{a} \times \mathbf{c}\).
Key Takeaway: The Cross Product produces a vector that sticks straight out of the surface formed by the first two vectors.
5. Geometric Applications of the Vector Product
One of the coolest things about the vector product is how it helps us find areas of shapes in 3D space.
Area of a Parallelogram
If you have a parallelogram with sides defined by vectors \(\mathbf{a}\) and \(\mathbf{b}\):
Area = \(|\mathbf{a} \times \mathbf{b}|\)
Area of a Triangle
Since a triangle is just half of a parallelogram:
Area = \(\frac{1}{2} |\mathbf{a} \times \mathbf{b}|\)
Alternative Line Equation
Using the property that the cross product of parallel vectors is zero, we can write a line equation as:
\((\mathbf{r} - \mathbf{a}) \times \mathbf{d} = \mathbf{0}\)
This says that the vector from our "anchor" point to any point on the line (\(\mathbf{r} - \mathbf{a}\)) must be parallel to the direction vector \(\mathbf{d}\).
Memory Aid:
Dot = \(\cos\) (used for angles).
Cross = \(\sin\) (used for areas).
Key Takeaway: The magnitude (length) of the cross product tells you the area of the shape formed by the two vectors.
Congratulations! You've covered the core concepts of Further Vectors. Keep practicing the calculations—once you master the determinant for the cross product, the rest is just logic!