Lesson 7.3: The Vector (Cross) Product and the Scalar Triple Product

Introduction

Welcome to Lesson 7.3 of Foundation Further Mathematics! In this lesson, we will explore two important concepts in vectors: the vector (cross) product and the scalar triple product.

Learning Objectives:

By the end of this lesson, students will be able to:

• Define the cross product and understand its geometric meaning.
• Calculate the magnitude of the cross product as the area of a parallelogram and determine its direction using the right-hand rule.
• Use the cross product to find a vector normal to two given vectors.
• Understand the scalar triple product $a \cdot (b \times c)$ as a measure of signed volume and apply it to a parallelepiped and tetrahedron.
• Utilize the scalar triple product to test whether three vectors are coplanar.

Let's get started! 🎉

The Cross Product

The cross product is a mathematical operation that takes two vectors and returns a new vector that is perpendicular to the plane formed by the original vectors. Let's denote two vectors as $\mathbf{a}$ and $\mathbf{b}$.

Definition and Geometric Meaning

The cross product of the vectors $\mathbf{a}$ and $\mathbf{b}$ is written as $\mathbf{a} \times \mathbf{b}$. The result is a vector $\mathbf{c}$:

$$\mathbf{c} = \mathbf{a} \times \mathbf{b}$$

Magnitude

The magnitude of the cross product can be calculated using the formula:

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

where $\theta$ is the angle between the two vectors. The magnitude represents the area of the parallelogram formed by the two vectors.

Direction

The direction of the resulting vector can be determined using the right-hand rule. To apply the right-hand rule:

1. Point your index finger in the direction of vector $\mathbf{a}$.
2. Point your middle finger in the direction of vector $\mathbf{b}$.
3. Your thumb will point in the direction of the cross product $\mathbf{c}$.

Example

Let's say we have the following vectors:

• $\mathbf{a} = (2, 3, 4)$
• $\mathbf{b} = (5, 6, 7)$

To calculate $\mathbf{c} = \mathbf{a} \times \mathbf{b}$:

$\mathbf{c} =egin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \\ 2 & 3 & 4 \\ 5 & 6 & 7\end{vmatrix} = (3 \cdot 7 - 4 \cdot 6)\mathbf{i} - (2 \cdot 7 - 4 \cdot 5)\mathbf{j} + (2 \cdot 6 - 3 \cdot 5)\mathbf{k}$

Calculating this gives us:

$$\mathbf{c} = (-10, 6, -3)$$

Vector Normal to Two Vectors

The cross product provides a straightforward means to find a vector normal to two other vectors. If $\mathbf{a}$ and $\mathbf{b}$ are not in the same direction, then $\mathbf{a} \times \mathbf{b}$ gives a vector that is perpendicular to both.

The Scalar Triple Product

The scalar triple product is a way to measure the signed volume of a parallelepiped formed by three vectors, $a$, $b$, and $c$. It is denoted as:

$$\text{Volume} = a \cdot (b \times c)$$

Meaning of the Scalar Triple Product

If the scalar triple product is zero, the three vectors are coplanar, meaning they lie on the same plane. If the value is positive, it indicates the orientation (handedness) of the volume defined by the vectors.

Example

Let’s consider vectors:

• $\mathbf{a} = (1, 1, 1)$
• $\mathbf{b} = (2, 3, 4)$
• $\mathbf{c} = (5, 6, 7)$

First, calculate $\mathbf{b} \times \mathbf{c}$:

$\mathbf{b} \times \mathbf{c} = egin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \\ 2 & 3 & 4 \\ 5 & 6 & 7\end{vmatrix}$

This yields:

$$\mathbf{b} \times \mathbf{c} = (-3, 6, -3)$$

Next, compute the scalar triple product:

$$\text{Volume} = a \cdot (b \times c) = (1, 1, 1) \cdot (-3, 6, -3) = -3 + 6 - 3 = 0$$

Since the volume is zero, it indicates that the vectors are coplanar!

Conclusion

In this lesson, students has learned about the vector cross product and the scalar triple product. We discovered how to compute the cross product of two vectors, its geometric meaning, and its applications to find normals and measure volume. These concepts are essential in understanding three-dimensional space and have implications in physics and engineering.

Study Notes

• The cross product $\mathbf{a} \times \mathbf{b}$ produces a vector that is perpendicular to $\mathbf{a}$ and $\mathbf{b}$.
• Magnitude of cross product: $|\mathbf{c}| = |\mathbf{a}||\mathbf{b}|\sin(\theta)$.
• Use the right-hand rule to determine the direction of the cross product.
• Scalar triple product $a \cdot (b \times c)$ gives signed volume of a parallelepiped.
• If scalar triple product is zero, vectors are coplanar.
• These concepts are useful in geometry, physics, and engineering! 🚀