Problem 14 - 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, \(|\mathbf{a} \times \mathbf{b}|\). $$ |\mathbf{a}|=4, \quad|\mathbf{b}|=5, \quad \theta=30^{\circ} $$

Step-by-Step Solution

Answer

|\mathbf{a} \times \mathbf{b}| = 10.

Step 1: Understanding the Cross Product

The cross product of two vectors \(\mathbf{a}\) and \(\mathbf{b}\) in three-dimensional space is another vector that is perpendicular to both \(\mathbf{a}\) and \(\mathbf{b}\). The magnitude of the cross product can be found using the formula: \[|\mathbf{a} \times \mathbf{b}| = |\mathbf{a}||\mathbf{b}|\sin(\theta)\], where \(\theta\) is the angle between the two vectors.

Step 2: Calculate the Sine of the Angle

Given \(\theta = 30^{\circ}\), we need to find \(\sin(30^{\circ})\). From trigonometry, we know that \(\sin(30^{\circ}) = \frac{1}{2}\).

Step 3: Apply the Cross Product Formula

Substitute the given lengths of the vectors and the sine of the angle into the cross product formula:\[|\mathbf{a} \times \mathbf{b}| = |4||5|\cdot \frac{1}{2}\].

Step 4: Perform the Multiplication

Calculate \[|\mathbf{a} \times \mathbf{b}| = 4 \cdot 5 \cdot \frac{1}{2} = 20 \cdot \frac{1}{2} = 10\].

Step 5: Conclusion and Verification

The magnitude of the cross product of vectors \(\mathbf{a}\) and \(\mathbf{b}\) is \(10\). You can verify this by considering the properties of the cross product, which confirm that the result is consistent with the given vector magnitudes and angle.

Key Concepts

Vector Magnitude

The magnitude of a vector is a measurement of its length or size. Imagine a straight arrow; the magnitude tells us how long this arrow is, without caring about its direction. A vector with a larger magnitude is like an arrow that is longer. In mathematical terms, for a vector \( \mathbf{v} = (v_1, v_2, v_3) \), its magnitude \( |\mathbf{v}| \) is calculated using the formula:\[|\mathbf{v}| = \sqrt{v_1^2 + v_2^2 + v_3^2}.\]This resembles the Pythagorean theorem in three dimensions.

• The magnitude gives us the "length" of the vector in space.
• For example, if \( \mathbf{a} = (x, y, z) \) with \( |\mathbf{a}| = 4 \), this means vector \( \mathbf{a} \) is 4 units long.
• Magnitude is always a non-negative number.

It simplifies finding how much of a specific quantity like time, distance, or force, the vector depicts in physics and engineering.

Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between side lengths and angles in triangles. It's fundamental when working with vectors, especially in geometry and physics.
When we say we need to use trigonometry to find \( \sin(30^{\circ}) \), we are discussing this exact application.

• The trigonometric function 'sine' (\( \sin \)) is particularly important for vector operations like the cross product, as it helps relate the angle between vectors to real-world calculations.
• Remember: \( \sin(30^{\circ}) = \frac{1}{2} \). This result doesn't depend on the vectors but solely on the angle 30°.
• Trigonometry links the spatial properties of vectors directly to calculations we can perform, such as computing the magnitude of a cross product.

Understanding these relationships makes solving vector problems much easier. It's like having a toolkit for turning geometrical relationships into numbers.

Angle Between Vectors

The angle between two vectors \( \mathbf{a} \) and \( \mathbf{b} \) is precisely the measure of separation between them. If you imagine the vectors as lines extending from a point, the angle is the space in between their directions.
To compute this angle, we employ mathematical operations that connect the geometry of vectors with arithmetic calculations.

• The angle, denoted \( \theta \), can be used to find the cross product magnitude: this involves \[ |\mathbf{a} \times \mathbf{b}| = |\mathbf{a}||\mathbf{b}|\sin(\theta). \] This formula tells us how these vectors interact in three-dimensional space.
• Angles can be measured in degrees or radians, but their function remains the same – defining the orientation difference between vectors.
• A smaller angle suggests the vectors are closer in direction, while a larger angle shows they point more broadly apart. In this case, the angle \( \theta = 30^{\circ} \) reflects a modest separation, relevant for many real-world vector operations.

With the angle, you can determine not just the vector orientations but also understand underlying relationships that define vector spaces, essential for everything from physics to engineering.