Problem 6 - Thomas Calculus

Question

In Exercises \(1-8,\) find the length and direction (when defined) of \(\mathbf{u} \times \mathbf{v}\) and \(\mathbf{v} \times \mathbf{u} .\) $$ \mathbf{u}=\mathbf{i} \times \mathbf{j}, \quad \mathbf{v}=\mathbf{j} \times \mathbf{k} $$

Step-by-Step Solution

Answer

The length of both cross products is 1; directions: \( \mathbf{j} \) and \(-\mathbf{j} \).

Step 1: Calculate \( \mathbf{u} \) and \( \mathbf{v} \)

Identify the cross products. Since \( \mathbf{u} = \mathbf{i} \times \mathbf{j} \), we know from the right-hand rule of cross products that \( \mathbf{u} = \mathbf{k} \). Similarly, for \( \mathbf{v} = \mathbf{j} \times \mathbf{k} \), we use the right-hand rule to find that \( \mathbf{v} = \mathbf{i} \).

Step 2: Find \( \mathbf{u} \times \mathbf{v} \)

We want to find the cross product \( \mathbf{u} \times \mathbf{v} \), which is \( \mathbf{k} \times \mathbf{i} \). Using the cyclic nature of the cross product, \( \mathbf{k} \times \mathbf{i} = \mathbf{j} \).

Step 3: Find \( \mathbf{v} \times \mathbf{u} \)

Now find \( \mathbf{v} \times \mathbf{u} \), which is \( \mathbf{i} \times \mathbf{k} \). Again, using the right-hand rule, we have \( \mathbf{i} \times \mathbf{k} = -\mathbf{j} \).

Step 4: Calculate the Length of Each Cross Product

The length (or magnitude) of a vector \( \mathbf{a} \) is given by \( |\mathbf{a}| \).\( |\mathbf{u} \times \mathbf{v}| = |\mathbf{j}| = 1 \)and \( |\mathbf{v} \times \mathbf{u}| = |-\mathbf{j}| = 1 \).

Step 5: Determine the Direction of Each Cross Product

The direction of \( \mathbf{u} \times \mathbf{v} \) is that of \( \mathbf{j} \), and the direction of \( \mathbf{v} \times \mathbf{u} \) is opposite to \( \mathbf{j} \).

Key Concepts

Right-Hand Rule

When working with the vector cross product, the right-hand rule is an essential tool to determine the direction of the resultant vector. To use the right-hand rule, follow these simple steps:

• Point your index finger in the direction of the first vector, \( \mathbf{u} \).
• Point your middle finger in the direction of the second vector, \( \mathbf{v} \).
• Your thumb will then point in the direction of the cross product, \( \mathbf{u} \times \mathbf{v} \).

This rule helps you visualize how the cross-product operation works and why the vector \( \mathbf{u} \times \mathbf{v} \) results in a new vector perpendicular to both \( \mathbf{u} \) and \( \mathbf{v} \). In our exercise, using the right-hand rule, we found that \( \mathbf{i} \times \mathbf{j} = \mathbf{k} \) and \( \mathbf{j} \times \mathbf{k} = \mathbf{i} \). It is crucial to remember that reversing the order of vectors, as in \( \mathbf{v} \times \mathbf{u} \), affects the direction, yielding \( -\mathbf{j} \).

Magnitude of Vectors

The magnitude of a vector refers to its length or size, which is a non-negative scalar. When dealing with cross products, the magnitude of the resultant vector can be calculated using:\[| \mathbf{u} \times \mathbf{v} | = |\mathbf{u}| |\mathbf{v}| \sin(\theta)\]where \( \theta \) is the angle between vectors \( \mathbf{u} \) and \( \mathbf{v} \). In three-dimensional space with unit vectors, the magnitude simplifies. For example, the magnitude of unit vectors \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) is 1. Thus:

• \(|\mathbf{u} \times \mathbf{v}| = |\mathbf{j}| = 1\)
• \(|\mathbf{v} \times \mathbf{u}| = |-\mathbf{j}| = 1\)

These magnitudes indicate that, despite the different directions of the resulting vectors, the magnitude remains the same. This property is consistent because the angle \( \theta \) is 90 degrees, making \( \sin(\theta) = 1 \).

Direction of Vectors

Direction is a key characteristic of vectors and is especially noteworthy in cross products. The cross product operation yields a vector that is orthogonal to the original vectors \( \mathbf{u} \) and \( \mathbf{v} \). The direction of this resultant vector depends on the right-hand rule, which we've covered previously.
In our specific example:

• \( \mathbf{u} \times \mathbf{v} = \mathbf{k} \times \mathbf{i} = \mathbf{j} \), so its direction is along the \( y \)-axis.
• \( \mathbf{v} \times \mathbf{u} = \mathbf{i} \times \mathbf{k} = -\mathbf{j} \), indicating an opposite direction along the \( y \)-axis.

Understanding vector direction in cross products is vital for visualizing the spatial relationships between vectors, reinforcing the concept of multidimensional vector mathematics, where orientation matters as much as magnitude.