Problem 15 - 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, \(|\mathbf{a} \times \mathbf{b}|\). $$|\mathbf{a}|=10, \quad|\mathbf{b}|=10, \quad \theta=90^{\circ}$$

Step-by-Step Solution

Answer

The length of the cross product is 100.

Step 1: Understanding the Cross Product Magnitude Formula

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

Step 2: Substituting Known Values

We know the magnitudes of \(\mathbf{a}\) and \(\mathbf{b}\) are both 10, and the angle \(\theta\) is \(90^{\circ}\). Substitute these values into the formula: \(|\mathbf{a} \times \mathbf{b}| = 10 \times 10 \times \sin(90^{\circ})\).

Step 3: Calculating Sine of 90 Degrees

The sine of a \(90^{\circ}\) angle is 1. So, \(\sin(90^{\circ}) = 1\).

Step 4: Calculate the Cross Product Magnitude

Replace \(\sin(90^{\circ})\) with 1 in the equation. So we have \(|\mathbf{a} \times \mathbf{b}| = 10 \times 10 \times 1\). Calculate the result to find \(|\mathbf{a} \times \mathbf{b}| = 100\).

Key Concepts

Vector Magnitude

When we talk about the magnitude of a vector, we are referring to its length or size. It's essential to understand how to calculate this as it forms part of operations with vectors, like the cross product. For a vector \( \mathbf{v} = (v_1, v_2, v_3) \), its magnitude is determined by the Pythagorean theorem in three dimensions.

• First, square each component of the vector: \( v_1^2, v_2^2, v_3^2 \).
• Then add these squares together: \( v_1^2 + v_2^2 + v_3^2 \).
• Finally, take the square root of the sum: \(|\mathbf{v}| = \sqrt{v_1^2 + v_2^2 + v_3^2}\).

This calculation gives you the magnitude, representing the vector's length in the geometric space. The magnitude is always a non-negative value.

Angle Between Vectors

The angle between two vectors, \( \mathbf{a} \) and \( \mathbf{b} \), is crucial when computing cross products. It influences not just the relationship between the vectors but also the magnitude of the resulting vector.
To find this angle mathematically, we can use the dot product formula. However, in cases involving the cross product, we can more directly use:

• Cross Product: \( \mathbf{a} \times \mathbf{b} = |\mathbf{a}||\mathbf{b}| \sin(\theta) \hat{n} \)
• Magnitude: Determined by both the magnitudes of the vectors and the sine of the angle between them.

For example, if the vectors are orthogonal (like in our exercise with \( \theta = 90^\circ \)), the cross product magnitude is simply \( |\mathbf{a}||\mathbf{b}| \), because \( \sin(90^\circ) = 1 \). This situation highlights both the importance of the angle and its role in maximizing the cross product's magnitude.

Sine Function

The sine function plays a vital role in calculating the magnitude of the cross product between two vectors. It's part of the trigonometric functions and relates specifically to right triangles. It defines the ratio of the length of the opposite side of an angle to its hypotenuse in a right triangle.
In vector mathematics, the sine of the angle between vectors helps determine how much of one vector points in the perpendicular direction of the other vector. Here's how it's useful:

• The sine function varies from -1 to 1, but when computing the magnitude, we always consider the absolute value (as it represents a distance).
• At \( \theta = 0^\circ \) or \( \theta = 180^\circ \), \( \sin(\theta) = 0 \), meaning vectors are parallel and the cross product is zero.
• When \( \theta = 90^\circ \), the sine value is 1, thus yielding the maximum possible magnitude of the cross product.

Understanding the sine function's behavior in these contexts helps clarify why certain angles produce stronger or weaker cross product magnitudes. In our example with \( \theta = 90^\circ \), the resultant magnitude illustrates how the two vectors are exactly perpendicular, maximizing the cross product.