Vector product and its applications (HL)

Introduction to the Vector Product

Welcome to one of the most powerful tools in your HL Mathematics toolkit! So far, you have likely encountered the scalar product (or dot product), which takes two vectors and spits out a single number. In this chapter, we explore the vector product (also known as the cross product).

The vector product is unique because when you "multiply" two vectors this way, the result is another vector. This new vector has some very special properties—most importantly, it is perfectly perpendicular (at a right angle) to the two original vectors. This makes it incredibly useful for finding distances and areas in 3D space.

1. Defining the Vector Product

For two vectors \(\mathbf{v}\) and \(\mathbf{w}\), the vector product is written as \(\mathbf{v} \times \mathbf{w}\). Unlike the scalar product, the order matters here!

The Magnitude (Size)

The magnitude of the vector product is defined by the formula:
\(|\mathbf{v} \times \mathbf{w}| = |\mathbf{v}| |\mathbf{w}| \sin \theta\)

Where \(\theta\) is the angle between the two vectors (ranging from \(0\) to \(\pi\)).

The Direction

The resulting vector \(\mathbf{v} \times \mathbf{w}\) is perpendicular to the plane containing \(\mathbf{v}\) and \(\mathbf{w}\). To figure out which way it points (up or down), we use the Right-Hand Rule:

• Point your index finger in the direction of vector \(\mathbf{v}\).
• Point your middle finger in the direction of vector \(\mathbf{w}\).
• Your thumb now points in the direction of \(\mathbf{v} \times \mathbf{w}\).

Quick Tip: If the two vectors are parallel, the angle \(\theta = 0\), and since \(\sin(0) = 0\), the vector product will be the zero vector, \(\mathbf{0}\).

2. Calculating the Vector Product (Component Form)

Don't worry if the trigonometry looks complicated; we usually calculate the vector product using the components of the vectors. If \(\mathbf{v} = \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix}\) and \(\mathbf{w} = \begin{pmatrix} w_1 \\ w_2 \\ w_3 \end{pmatrix}\), the formula is:

\(\mathbf{v} \times \mathbf{w} = \begin{pmatrix} v_2w_3 - v_3w_2 \\ v_3w_1 - v_1w_3 \\ v_1w_2 - v_2w_1 \end{pmatrix}\)

How to remember this without memorizing:

Many students prefer the "determinant method." Write the unit vectors \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\) in the top row, and the components of the vectors in the rows below:

\(\mathbf{v} \times \mathbf{w} = \det \begin{pmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ v_1 & v_2 & v_3 \\ w_1 & w_2 & w_3 \end{pmatrix}\)

Step-by-Step Calculation:
1. For the \(\mathbf{i}\) component: Ignore the \(\mathbf{i}\) column and cross-multiply the remaining four numbers: \((v_2 \times w_3) - (v_3 \times w_2)\).
2. For the \(\mathbf{j}\) component: Crucial Step! Put a minus sign in front, ignore the \(\mathbf{j}\) column: \(-(v_1 \times w_3 - v_3 \times w_1)\).
3. For the \(\mathbf{k}\) component: Ignore the \(\mathbf{k}\) column: \((v_1 \times w_2 - v_2 \times w_1)\).

Example: If \(\mathbf{v} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}\) and \(\mathbf{w} = \begin{pmatrix} 4 \\ 5 \\ 6 \end{pmatrix}\), then:
\(\mathbf{v} \times \mathbf{w} = \begin{pmatrix} (2 \times 6) - (3 \times 5) \\ -( (1 \times 6) - (3 \times 4) ) \\ (1 \times 5) - (2 \times 4) \end{pmatrix} = \begin{pmatrix} 12-15 \\ -(6-12) \\ 5-8 \end{pmatrix} = \begin{pmatrix} -3 \\ 6 \\ -3 \end{pmatrix}\).

3. Properties of the Vector Product

It is important to know how the vector product behaves. It doesn't always act like normal multiplication!

• Anti-commutative: \(\mathbf{v} \times \mathbf{w} = -(\mathbf{w} \times \mathbf{v})\). If you swap the order, you flip the direction!
• Distributive: \(\mathbf{u} \times (\mathbf{v} + \mathbf{w}) = \mathbf{u} \times \mathbf{v} + \mathbf{u} \times \mathbf{w}\).
• Parallel Vectors: \(\mathbf{v} \times \mathbf{v} = \mathbf{0}\) (Any vector "crossed" with itself is zero).
• Scalar Multiplication: \((k\mathbf{v}) \times \mathbf{w} = k(\mathbf{v} \times \mathbf{w})\).

Key Takeaway: The most useful property is that \(\mathbf{v} \times \mathbf{w}\) is orthogonal (perpendicular) to both \(\mathbf{v}\) and \(\mathbf{w}\). You can check this by showing that \((\mathbf{v} \times \mathbf{w}) \cdot \mathbf{v} = 0\).

4. Geometric Applications

The vector product isn't just an abstract calculation; it represents physical space.

Area of a Parallelogram

If you have two vectors \(\mathbf{v}\) and \(\mathbf{w}\) starting from the same point, they form two sides of a parallelogram. The area of that parallelogram is simply the magnitude of their vector product:

Area of Parallelogram = \(|\mathbf{v} \times \mathbf{w}|\)

Area of a Triangle

Since a triangle is exactly half of a parallelogram, if a triangle has sides defined by vectors \(\mathbf{v}\) and \(\mathbf{w}\):

Area of Triangle = \(\frac{1}{2} |\mathbf{v} \times \mathbf{w}|\)

Common Exam Task: You are given three points \(A\), \(B\), and \(C\). To find the area of triangle \(ABC\):
1. Find the displacement vectors \(\vec{AB}\) and \(\vec{AC}\).
2. Calculate the vector product \(\vec{AB} \times \vec{AC}\).
3. Find the magnitude of this result and divide by \(2\).

5. Summary and Common Pitfalls

Quick Review:
- Use the vector product when you need a vector perpendicular to two others.
- Use the magnitude \(|\mathbf{v} \times \mathbf{w}|\) to find areas.
- Remember that \(\mathbf{v} \times \mathbf{w}\) results in a vector, whereas \(\mathbf{v} \cdot \mathbf{w}\) results in a number.

Common Mistakes to Avoid:
- Forgetting the minus sign on the middle (\(\mathbf{j}\)) component: This is the most common calculation error in IB exams!
- Mixing up Sine and Cosine: Remember, Dot product uses \(\cos \theta\), Cross product uses \(\sin \theta\).
- Confusing the result: \(\mathbf{v} \times \mathbf{w}\) is a vector. If you are asked for the area, you must find the magnitude of that vector. Don't leave your final answer for area as a vector!

Note: In the next chapters, you will see how the vector product is used to find the equations of planes (AHL 3.17) by providing the "normal vector" to the surface.