Problem 16 - Precalculus Mathematics for Calculus

Question

The lengths of two vectors a and \(b\) and the angle \(\theta\) between them are given. Find the length of their cross product, \(|\mathbf{a} \times \mathbf{b}|\). $$|\mathbf{a}|=0.12, \quad|\mathbf{b}|=1.25, \quad \theta=75^{\circ}$$

Step-by-Step Solution

Answer

The magnitude of the cross product is approximately 0.1449.

Step 1: Understand the Cross Product Formula

The magnitude of the cross product of two vectors \(\mathbf{a}\) and \(\mathbf{b}\) is given by the formula \(|\mathbf{a} \times \mathbf{b}| = |\mathbf{a}| \cdot |\mathbf{b}| \cdot \sin(\theta)\), where \(|\mathbf{a}|\) and \(|\mathbf{b}|\) are the magnitudes of the vectors, and \(\theta\) is the angle between them.

Step 2: Convert Angle

Convert the angle \(\theta = 75^{\circ}\) to radians if necessary, but since the sine function can take degrees directly in many calculators, check your calculator settings. Otherwise, use the conversion: \(\theta_{rad} = \frac{75 \cdot \pi}{180}\).

Step 3: Calculate Sine of Theta

Calculate \(\sin(75^{\circ})\) using a calculator. The sine of \(75^{\circ}\) is approximately \(0.9659\).

Step 4: Plug Values into Cross Product Formula and Calculate

Now use the formula \(|\mathbf{a} \times \mathbf{b}| = |\mathbf{a}| \cdot |\mathbf{b}| \cdot \sin(\theta)\). Substitute \(|\mathbf{a}| = 0.12\), \(|\mathbf{b}| = 1.25\), and \(\sin(75^{\circ}) = 0.9659\) into the equation:\[|\mathbf{a} \times \mathbf{b}| = 0.12 \cdot 1.25 \cdot 0.9659.\]Calculate this expression to find the magnitude of the cross product.

Step 5: Final Calculation

Calculate the product: \[0.12 \cdot 1.25 \cdot 0.9659 = 0.144885\]Thus, the magnitude of the cross product is approximately \(0.1449\).

Key Concepts

Vector Magnitude

The magnitude of a vector is a measure of its length. Imagine a vector as an arrow pointing from one point to another in space. The magnitude corresponds to the length of this arrow. It provides a scalar (a single number) to represent the vector's size, without considering its direction. To calculate the magnitude of a vector \( \mathbf{a} \) with components \((a_1, a_2, a_3)\), you can use the formula:

• \( |\mathbf{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2} \)

In our exercise, the magnitude of vector \(\mathbf{a}\) is given as \(0.12\), and that of \(\mathbf{b}\) is \(1.25\). These values are essential when using the cross product formula.

Sine Function

The sine function is a fundamental trigonometric function that appears in various math and physics problems. It relates the angle of a right triangle to the ratio of the length of the opposite side to the hypotenuse.For any angle \( \theta \) in degrees, the sine function is written as \( \sin(\theta) \). In our problem, we use \( \sin(75^\circ) \), which calculates to approximately \( 0.9659 \).

• The sine function is periodic, meaning it repeats its values in cycles.
• It ranges between \(-1\) and \(1\).

The sine value is crucial in the cross product formula, as it helps determine the projection of the vectors' interaction concerning the angle between them.

Angle Conversion

Angle conversion is the process of changing an angle from degrees to radians or vice versa. The standard unit of angular measure in mathematics is the radian. However, angles are often measured in degrees in practical problems, so conversion is common.To convert an angle from degrees to radians, use the formula:

• \( \theta_{rad} = \theta_{deg} \cdot \frac{\pi}{180} \)

In this task, \(75^\circ\) can be converted to radians as \( \frac{75 \cdot \pi}{180} \). However, since many calculators can directly process the sine function in degrees, knowing how to convert might not always be necessary unless specified otherwise.

Trigonometry

Trigonometry is the study of the relationships between the sides and angles of triangles. It is fundamental in understanding not just triangle properties but also periodic patterns and wave phenomena in physics and engineering. The cross product formula \( |\mathbf{a} \times \mathbf{b}| = |\mathbf{a}| \cdot |\mathbf{b}| \cdot \sin(\theta) \) is directly connected to trigonometric concepts.

• The formula shows how magnitudes and the sine of the angle between vectors affect their cross product.
• Understanding these relationships highlights how vectors combine in space to form new, perpendicular vectors.

Trigonometry empowers you to transform between visual geometric interpretations and mathematical expressions directly.