Vectors

Q: How are the component and geometric forms of the scalar (dot) product defined?

\(\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 = |\mathbf{a}||\mathbf{b}|\cos\theta\), where \(\theta\) is the angle between the two vectors.

Q: How can the scalar product be used to find the angle between two vectors and test if they are perpendicular?

The angle is given by \(\cos\theta = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}||\mathbf{b}|}\); two non-zero vectors are perpendicular if and only if \(\mathbf{a} \cdot \mathbf{b} = 0\).

Q: How do you calculate the vector (cross) product \(\mathbf{a} \times \mathbf{b}\)?

Evaluate the \(3 \times 3\) determinant \(\mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix} = \begin{pmatrix} a_2b_3 - a_3b_2 \\ -(a_1b_3 - a_3b_1) \\ a_1b_2 - a_2b_1 \end{pmatrix}\).

Q: What is the geometric magnitude of the vector product, and how does it test for parallel vectors?

\(|\mathbf{a} \times \mathbf{b}| = |\mathbf{a}||\mathbf{b}|\sin\theta\); two non-zero vectors are parallel if and only if \(\mathbf{a} \times \mathbf{b} = \mathbf{0}\).

Q: How is the cross product used to find the area of a parallelogram or triangle in 3D?

For adjacent edge vectors \(\mathbf{a}\) and \(\mathbf{b}\), \(\text{Area of Parallelogram} = |\mathbf{a} \times \mathbf{b}|\) and \(\text{Area of Triangle} = \frac{1}{2}|\mathbf{a} \times \mathbf{b}|\).

Q: What are the vector, parametric, and Cartesian forms of a straight line in 3D?

Vector: \(\mathbf{r} = \mathbf{a} + \lambda\mathbf{d}\); Parametric: \(x = a_1 + \lambda d_1\), \(y = a_2 + \lambda d_2\), \(z = a_3 + \lambda d_3\); Cartesian: \(\frac{x - a_1}{d_1} = \frac{y - a_2}{d_2} = \frac{z - a_3}{d_3}\).

Q: How do you calculate the acute angle between two straight lines in 3D?

Use the dot product of their direction vectors: \(\cos\theta = \frac{|\mathbf{d}_1 \cdot \mathbf{d}_2|}{|\mathbf{d}_1||\mathbf{d}_2|}\).

Q: What are skew lines in 3D space?

Skew lines are lines in 3D space that are not parallel to each other and do not intersect.

Q: What is the step-by-step method to determine if two lines intersect or are skew?

Check if direction vectors are parallel; if not, equate component equations using distinct parameters \(\lambda\) and \(\mu\), solve two equations, and test the solution in the third equation.

Further Mathematics 2330 · CCEA AS Level · 11 min read · Updated: 31 Aug 2026

Welcome to 3D Vectors!

Welcome to one of the most visual and exciting chapters in AS 1: Pure Mathematics! In standard A-Level Maths, you met vectors in two dimensions (flat surfaces) and touched on basic three-dimensional coordinates. In Further Mathematics, we take things to the next level by unlocking powerful tools in 3D space: the scalar product, the vector product, and the geometry of 3D straight lines.

Did you know? The exact vector mathematics you are about to learn is what powers modern 3D video game engines (like Unreal Engine and Unity), flight simulators, and aerospace navigation! Don't worry if three dimensions feel tricky to picture at first—we will break every single idea down step-by-step with clear recipes and analogies.

1. Vectors Refresher & The Scalar (Dot) Product

A vector has both a magnitude (size) and a direction. In 3D space, we write a vector using base unit vectors \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\) (which point along the positive \(x\), \(y\), and \(z\) axes respectively) or as a column vector:

\(\mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} + a_3\mathbf{k} = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix}\)

The magnitude (or length) of vector \(\mathbf{a}\) is found using 3D Pythagoras:

\(|\mathbf{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2}\)

What is the Scalar (Dot) Product?

The scalar product (often called the dot product) is a way of multiplying two vectors together to get a single scalar number (not a vector!).

Memory Trick: Dot product results in a Directionless number (a scalar).

There are two essential definitions for the dot product between \(\mathbf{a}\) and \(\mathbf{b}\):

1. Component Form: Multiply matching components and add them together:
\(\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3\)

2. Geometric Form: Using lengths and the angle \(\theta\) between them:
\(\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}||\mathbf{b}|\cos\theta\)

Finding the Angle Between Two Vectors

By rearranging the geometric formula, we can find the angle \(\theta\) between any two non-zero vectors:

\(\cos\theta = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}||\mathbf{b}|} = \frac{a_1b_1 + a_2b_2 + a_3b_3}{\sqrt{a_1^2 + a_2^2 + a_3^2}\sqrt{b_1^2 + b_2^2 + b_3^2}}\)

The Perpendicular Test

If two non-zero vectors are perpendicular (at right angles, \(\theta = 90^\circ\)), then \(\cos 90^\circ = 0\). This gives us a crucial rule:

Two non-zero vectors \(\mathbf{a}\) and \(\mathbf{b}\) are perpendicular if and only if:
\(\mathbf{a} \cdot \mathbf{b} = 0\)

Step-by-Step Example: Finding the Angle

Find the angle between \(\mathbf{a} = 2\mathbf{i} - \mathbf{j} + 2\mathbf{k}\) and \(\mathbf{b} = 4\mathbf{i} + 0\mathbf{j} - 3\mathbf{k}\).

Step 1: Calculate the dot product \(\mathbf{a} \cdot \mathbf{b}\):
\(\mathbf{a} \cdot \mathbf{b} = (2)(4) + (-1)(0) + (2)(-3) = 8 + 0 - 6 = 2\)

Step 2: Calculate the magnitude of each vector:
\(|\mathbf{a}| = \sqrt{2^2 + (-1)^2 + 2^2} = \sqrt{4 + 1 + 4} = \sqrt{9} = 3\)
\(|\mathbf{b}| = \sqrt{4^2 + 0^2 + (-3)^2} = \sqrt{16 + 0 + 9} = \sqrt{25} = 5\)

Step 3: Substitute into the angle formula:
\(\cos\theta = \frac{2}{3 \times 5} = \frac{2}{15}\)
\(\theta = \arccos\left(\frac{2}{15}\right) \approx 82.3^\circ\)

Key Takeaway for Section 1: The scalar product gives a real number. If \(\mathbf{a} \cdot \mathbf{b} = 0\), the vectors are perpendicular. If \(\mathbf{a} \cdot \mathbf{b} > 0\), the angle is acute; if \(\mathbf{a} \cdot \mathbf{b} < 0\), the angle is obtuse.

2. The Vector (Cross) Product

Unlike the scalar product, the vector product (also called the cross product, written \(\mathbf{a} \times \mathbf{b}\)) produces a new vector that is perpendicular to both original vectors \(\mathbf{a}\) and \(\mathbf{b}\).

Everyday Analogy: Imagine using a screwdriver. You turn it along vector \(\mathbf{a}\) towards vector \(\mathbf{b}\), and the screw drives directly outwards along \(\mathbf{a} \times \mathbf{b}\) (perpendicular to your turning plane)!

Calculating the Cross Product (The Determinant Method)

To calculate \(\mathbf{a} \times \mathbf{b}\) where \(\mathbf{a} = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} b_1 \\ b_2 \\ b_3 \end{pmatrix}\), set up a \(3 \times 3\) determinant:

\(\mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}\)

Expanding along the top row (remembering the middle sign is negative):

\(\mathbf{a} \times \mathbf{b} = \mathbf{i}(a_2b_3 - a_3b_2) - \mathbf{j}(a_1b_3 - a_3b_1) + \mathbf{k}(a_1b_2 - a_2b_1)\)

In column form, this is:

\(\mathbf{a} \times \mathbf{b} = \begin{pmatrix} a_2b_3 - a_3b_2 \\ -(a_1b_3 - a_3b_1) \\ a_1b_2 - a_2b_1 \end{pmatrix} = \begin{pmatrix} a_2b_3 - a_3b_2 \\ a_3b_1 - a_1b_3 \\ a_1b_2 - a_2b_1 \end{pmatrix}\)

Geometric Magnitude & Parallel Vectors

The magnitude of the cross product is given by:

\(|\mathbf{a} \times \mathbf{b}| = |\mathbf{a}||\mathbf{b}|\sin\theta\)

where \(\theta\) is the angle between \(\mathbf{a}\) and \(\mathbf{b}\) (\(0^\circ \le \theta \le 180^\circ\)).

Important Properties:
• Anti-commutative: \(\mathbf{b} \times \mathbf{a} = -(\mathbf{a} \times \mathbf{b})\). Swapping the order flips the vector direction 180 degrees!
• Parallel Test: If two non-zero vectors are parallel, \(\theta = 0^\circ\) or \(180^\circ\), so \(\sin\theta = 0\). Therefore:
\(\mathbf{a} \text{ and } \mathbf{b} \text{ are parallel} \iff \mathbf{a} \times \mathbf{b} = \mathbf{0}\) (the zero vector \(\begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}\)).
• \(\mathbf{a} \times \mathbf{a} = \mathbf{0}\) for any vector \(\mathbf{a}\).

Geometric Applications: Areas

The vector product makes finding areas in 3D remarkably easy:

1. Area of a Parallelogram with adjacent sides \(\mathbf{a}\) and \(\mathbf{b}\):
\(\text{Area} = |\mathbf{a} \times \mathbf{b}|\)

2. Area of a Triangle with adjacent sides \(\mathbf{a}\) and \(\mathbf{b}\):
\(\text{Area} = \frac{1}{2}|\mathbf{a} \times \mathbf{b}|\)

Step-by-Step Example: Area of a Triangle

Find the area of the triangle with vertices \(A(1, 0, 2)\), \(B(2, 3, 1)\), and \(C(0, 1, 4)\).

Step 1: Find two edge vectors sharing vertex \(A\):
\(\vec{AB} = \mathbf{b} - \mathbf{a} = \begin{pmatrix} 2 - 1 \\ 3 - 0 \\ 1 - 2 \end{pmatrix} = \begin{pmatrix} 1 \\ 3 \\ -1 \end{pmatrix}\)
\(\vec{AC} = \mathbf{c} - \mathbf{a} = \begin{pmatrix} 0 - 1 \\ 1 - 0 \\ 4 - 2 \end{pmatrix} = \begin{pmatrix} -1 \\ 1 \\ 2 \end{pmatrix}\)

Step 2: Calculate the cross product \(\vec{AB} \times \vec{AC}\):
\(\vec{AB} \times \vec{AC} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & 3 & -1 \\ -1 & 1 & 2 \end{vmatrix}\)
\(\mathbf{i}\text{-component: } (3)(2) - (-1)(1) = 6 - (-1) = 7\)
\(\mathbf{j}\text{-component: } -((1)(2) - (-1)(-1)) = -(2 - 1) = -1\)
\(\mathbf{k}\text{-component: } (1)(1) - (3)(-1) = 1 - (-3) = 4\)
So, \(\vec{AB} \times \vec{AC} = 7\mathbf{i} - \mathbf{j} + 4\mathbf{k} = \begin{pmatrix} 7 \\ -1 \\ 4 \end{pmatrix}\).

Step 3: Find the magnitude and halve it:
\(|\vec{AB} \times \vec{AC}| = \sqrt{7^2 + (-1)^2 + 4^2} = \sqrt{49 + 1 + 16} = \sqrt{66}\)
\(\text{Area} = \frac{1}{2}\sqrt{66} \approx 4.06 \text{ square units}\)

Key Takeaway for Section 2: The cross product produces a vector perpendicular to both input vectors. Its magnitude equals the area of the parallelogram formed by the two vectors.

3. Straight Lines in 3D

To define a straight line in 3D, you need two pieces of information:
1. A known fixed point on the line (given by position vector \(\mathbf{a}\)).
2. A direction vector along which the line travels (given by \(\mathbf{d}\)).

Forms of a 3D Line Equation

1. Vector Form:
\(\mathbf{r} = \mathbf{a} + \lambda\mathbf{d}\)
where \(\mathbf{r} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}\) is the position vector of any general point on the line, and \(\lambda\) is a scalar parameter (like a time or distance slider).

2. Parametric Form:
Writing each coordinate separately:
\(x = a_1 + \lambda d_1\)
\(y = a_2 + \lambda d_2\)
\(z = a_3 + \lambda d_3\)

3. Cartesian Form:
Rearranging each parametric equation for \(\lambda\) and equating them:
\(\lambda = \frac{x - a_1}{d_1} = \frac{y - a_2}{d_2} = \frac{z - a_3}{d_3}\)
So the Cartesian equation is written as:
\(\frac{x - a_1}{d_1} = \frac{y - a_2}{d_2} = \frac{z - a_3}{d_3}\)

Watch out for zeroes! If one of the direction components is zero (e.g., \(d_2 = 0\)), we write:
\(\frac{x - a_1}{d_1} = \frac{z - a_3}{d_3}, \quad y = a_2\)

Angle Between Two Lines

The angle \(\theta\) between two lines is simply the angle between their direction vectors \(\mathbf{d}_1\) and \(\mathbf{d}_2\). We find the acute angle using the dot product formula:

\(\cos\theta = \frac{|\mathbf{d}_1 \cdot \mathbf{d}_2|}{|\mathbf{d}_1||\mathbf{d}_2|}\)

Key Takeaway for Section 3: A line is fully defined by \(\mathbf{r} = \mathbf{a} + \lambda\mathbf{d}\). The point \(\mathbf{a}\) anchors the line, while the direction vector \(\mathbf{d}\) controls its angle in space.

4. Relationships Between Lines in 3D: Intersecting, Parallel, or Skew

In 2D geometry, two lines must either be parallel or they must intersect. But in 3D, there is a third fascinating possibility: skew lines!

Real-World Analogy for Skew Lines: Think of a motorway flyover. The road on the bridge goes east-west, and the road below goes north-south. They are not parallel, yet they never cross or crash! Lines in 3D that are not parallel and never intersect are called skew.

Summary of Possibilities for Two Lines \(L_1\) and \(L_2\):

• Parallel: Direction vectors \(\mathbf{d}_1\) and \(\mathbf{d}_2\) are scalar multiples of each other (\(\mathbf{d}_1 = k\mathbf{d}_2\)).
• Intersecting: There is a unique point that lies on both lines (direction vectors are not parallel, and equations have a consistent solution).
• Skew: Direction vectors are not parallel, and no common intersection point exists.

Method: How to Test for Intersection or Skew

Let line \(L_1\) be \(\mathbf{r}_1 = \mathbf{a}_1 + \lambda\mathbf{d}_1\) and line \(L_2\) be \(\mathbf{r}_2 = \mathbf{a}_2 + \mu\mathbf{d}_2\). (Always use two different letters, like \(\lambda\) and \(\mu\), for the parameters!)

Step 1: Check direction vectors. If \(\mathbf{d}_1\) is a multiple of \(\mathbf{d}_2\), the lines are parallel. If not, proceed to Step 2.
Step 2: Equate \(x\), \(y\), and \(z\) components:
\((1) \quad x_1 + \lambda d_{1x} = x_2 + \mu d_{2x}\)
\((2) \quad y_1 + \lambda d_{1y} = y_2 + \mu d_{2y}\)
\((3) \quad z_1 + \lambda d_{1z} = z_2 + \mu d_{2z}\)
Step 3: Solve equations (1) and (2) simultaneously to find values for \(\lambda\) and \(\mu\).
Step 4: Test in equation (3):
• If \(\lambda\) and \(\mu\) satisfy equation (3), the lines intersect! Substitute \(\lambda\) back into \(L_1\) to get the coordinates of the intersection point.
• If \(\lambda\) and \(\mu\) do not satisfy equation (3), the lines are skew.

Step-by-Step Example: Testing Lines

Determine whether the following lines intersect, are parallel, or are skew:

\(L_1: \mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix} + \lambda\begin{pmatrix} 1 \\ -1 \\ 2 \end{pmatrix}\) and \(L_2: \mathbf{r} = \begin{pmatrix} 4 \\ -1 \\ 1 \end{pmatrix} + \mu\begin{pmatrix} 2 \\ 1 \\ -1 \end{pmatrix}\)

Step 1: Check for parallelism:
\(\mathbf{d}_1 = \begin{pmatrix} 1 \\ -1 \\ 2 \end{pmatrix}\) is not a scalar multiple of \(\mathbf{d}_2 = \begin{pmatrix} 2 \\ 1 \\ -1 \end{pmatrix}\). So they are not parallel.

Step 2: Set up 3 equations by equating components:
\((1) \quad 1 + \lambda = 4 + 2\mu \implies \lambda - 2\mu = 3\)
\((2) \quad 2 - \lambda = -1 + \mu \implies -\lambda - \mu = -3 \implies \lambda + \mu = 3\)
\((3) \quad -1 + 2\lambda = 1 - \mu \implies 2\lambda + \mu = 2\)

Step 3: Solve (1) and (2):
Adding (1) and (2):
\((\lambda - 2\mu) + (\lambda + \mu) = 3 + 3\)
\(2\lambda - \mu = 6\)
From (2), \(\lambda = 3 - \mu\). Substitute into (1):
\((3 - \mu) - 2\mu = 3 \implies 3 - 3\mu = 3 \implies -3\mu = 0 \implies \mu = 0\)
Then \(\lambda = 3 - 0 = 3\).

Step 4: Test in (3):
Left side of (3): \(2(3) + 0 = 6\)
Right side of (3): \(2\)
Since \(6 \ne 2\), the third equation is not satisfied.

Conclusion: The lines do not intersect. Since they are also not parallel, they are skew.

Common Mistakes to Avoid

• Mixing up Dot and Cross Product Outputs: Remember that \(\mathbf{a} \cdot \mathbf{b}\) is a scalar (a number), while \(\mathbf{a} \times \mathbf{b}\) is a 3D vector.
• Using the Same Parameter Letter: When setting up two lines to find an intersection, always use \(\lambda\) for line 1 and \(\mu\) for line 2. If you use \(\lambda\) for both, you assume they reach the intersection at the same "time", which is usually not true!
• Sign Error in Cross Product: Don't forget the negative sign on the middle term when expanding determinants: \(\mathbf{i}(\dots) \mathbf{-} \mathbf{j}(\dots) + \mathbf{k}(\dots)\).
• Forgetting to Check the Third Equation: Always substitute your \(\lambda\) and \(\mu\) back into the 3rd equation to confirm whether lines truly intersect or are skew.

Quick Reference Summary

Scalar Product: \(\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 = |\mathbf{a}||\mathbf{b}|\cos\theta\)
Perpendicular Condition: \(\mathbf{a} \cdot \mathbf{b} = 0\)
Vector Product: \(\mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}\)
Parallel Condition: \(\mathbf{a} \times \mathbf{b} = \mathbf{0}\)
Area of Triangle: \(\text{Area} = \frac{1}{2}|\mathbf{a} \times \mathbf{b}|\)
Area of Parallelogram: \(\text{Area} = |\mathbf{a} \times \mathbf{b}|\)
Vector Line: \(\mathbf{r} = \mathbf{a} + \lambda\mathbf{d}\)
Cartesian Line: \(\frac{x - a_1}{d_1} = \frac{y - a_2}{d_2} = \frac{z - a_3}{d_3}\)

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

How are the component and geometric forms of the scalar (dot) product defined?

\(\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 = |\mathbf{a}||\mathbf{b}|\cos\theta\), where \(\theta\) is the angle between the two vectors.

How can the scalar product be used to find the angle between two vectors and test if they are perpendicular?

The angle is given by \(\cos\theta = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}||\mathbf{b}|}\); two non-zero vectors are perpendicular if and only if \(\mathbf{a} \cdot \mathbf{b} = 0\).

How do you calculate the vector (cross) product \(\mathbf{a} \times \mathbf{b}\)?

Evaluate the \(3 \times 3\) determinant \(\mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix} = \begin{pmatrix} a_2b_3 - a_3b_2 \\ -(a_1b_3 - a_3b_1) \\ a_1b_2 - a_2b_1 \end{pmatrix}\).

What is the geometric magnitude of the vector product, and how does it test for parallel vectors?

\(|\mathbf{a} \times \mathbf{b}| = |\mathbf{a}||\mathbf{b}|\sin\theta\); two non-zero vectors are parallel if and only if \(\mathbf{a} \times \mathbf{b} = \mathbf{0}\).

How is the cross product used to find the area of a parallelogram or triangle in 3D?

For adjacent edge vectors \(\mathbf{a}\) and \(\mathbf{b}\), \(\text{Area of Parallelogram} = |\mathbf{a} \times \mathbf{b}|\) and \(\text{Area of Triangle} = \frac{1}{2}|\mathbf{a} \times \mathbf{b}|\).

What are the vector, parametric, and Cartesian forms of a straight line in 3D?

Vector: \(\mathbf{r} = \mathbf{a} + \lambda\mathbf{d}\); Parametric: \(x = a_1 + \lambda d_1\), \(y = a_2 + \lambda d_2\), \(z = a_3 + \lambda d_3\); Cartesian: \(\frac{x - a_1}{d_1} = \frac{y - a_2}{d_2} = \frac{z - a_3}{d_3}\).

How do you calculate the acute angle between two straight lines in 3D?

Use the dot product of their direction vectors: \(\cos\theta = \frac{|\mathbf{d}_1 \cdot \mathbf{d}_2|}{|\mathbf{d}_1||\mathbf{d}_2|}\).

What are skew lines in 3D space?

Skew lines are lines in 3D space that are not parallel to each other and do not intersect.

What is the step-by-step method to determine if two lines intersect or are skew?

Check if direction vectors are parallel; if not, equate component equations using distinct parameters \(\lambda\) and \(\mu\), solve two equations, and test the solution in the third equation.

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