Problem 15 - Algebra and Trigonometry

Question

The lengths of two vectors \(\mathbf{a}\) and \(\mathbf{b}\) and the angle \(\theta\) between them are given. Find the length of their cross product, \(|\mathbf{a} \times \mathbf{b}|\). $$ |\mathbf{a}|=10, \quad|\mathbf{b}|=10, \quad \theta=90^{\circ} $$

Step-by-Step Solution

Answer

The magnitude of the cross product is 100.

Step 1: Understand the Problem

We need to find the magnitude of the cross product of two vectors \( \mathbf{a} \) and \( \mathbf{b} \). We are given the magnitudes of these vectors and the angle between them.

Step 2: Recall the Formula for Cross Product Magnitude

The magnitude of the cross product \( |\mathbf{a} \times \mathbf{b}| \) is given by the formula: \[ |\mathbf{a} \times \mathbf{b}| = |\mathbf{a}| \cdot |\mathbf{b}| \cdot \sin(\theta) \] where \( \theta \) is the angle between the two vectors.

Step 3: Substitute the Given Values

Substitute the given values into the formula: \( |\mathbf{a}| = 10 \), \( |\mathbf{b}| = 10 \), and \( \theta = 90^{\circ} \). So the formula becomes: \[ |\mathbf{a} \times \mathbf{b}| = 10 \cdot 10 \cdot \sin(90^{\circ}) \]

Step 4: Calculate \(\sin(90^{\circ})\)

The sine of \( 90^{\circ} \) is 1. Therefore, the expression becomes: \[ |\mathbf{a} \times \mathbf{b}| = 10 \cdot 10 \cdot 1 \]

Step 5: Compute the Result

Now, perform the multiplication: \[ |\mathbf{a} \times \mathbf{b}| = 100 \] The magnitude of the cross product is 100.

Key Concepts

Vectors

Vectors are mathematical entities characterized not only by their magnitude, or size, but also by their direction. They are crucial in understanding and describing physical quantities such as force, velocity, and displacement, which have both of these attributes.

A vector can be represented in a coordinate system using coordinates which indicate its position relative to a specific point. For instance, a vector \( \mathbf{a} \) in three dimensions might be denoted as \( \mathbf{a} = (a_1, a_2, a_3) \), where each component is akin to its influence in the respective x, y, z axes.

Vectors can be added together and scaled, which means multiplying them by a scalar. These operations follow specific rules. Importantly, vectors can be multiplied in two specific ways: the dot product and the cross product. While the dot product results in a scalar and uses the angle's cosine, the cross product's outcome is a vector and involves the sine of the angle.

Magnitude

The magnitude of a vector refers to its length or size and gives us an idea of the distance it covers in space without considering its direction.

To find the magnitude of a vector \( \mathbf{a} = (a_1, a_2, a_3) \), you use the formula:

• \[|\mathbf{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2}\]

The magnitude is always a non-negative number, which, in the context of this exercise, is given directly. This makes it simpler as calculations start with readily known values.

It's worth noting that when vectors are cross-multiplied, the resultant vector's magnitude tells you the area of the parallelogram spanned by the original vectors. This makes the computation of magnitudes significant in geometry and physics, where space and arrangement are key considerations.

Angle between Vectors

The angle between two vectors is an important factor in many vector operations, including cross and dot products.

When we talk about the angle \(\theta\) between two vectors, we mean the smallest angle that would change one vector into the other if they were placed tail-to-tail. In this setup:

• The dot product is influenced by the cosine of \(\theta\).
• The cross product relies on the sine of \(\theta\).

Given the exercise, the angle is provided as 90 degrees. This is a favorable scenario since \(\sin(90^{\circ}) = 1\), simplifying calculations when figuring out the cross product magnitude.

Understanding how angles affect vector interactions is crucial. A \(90^{\circ}\) angle makes vectors orthogonal, meaning they face perpendicular directions. This leads to maximum cross product magnitude, making angles a pivotal concept in spatial vector analysis.