Problem 14 - Calculus Early Transcendentals: Pearson New International Edition

Question

If \(\mathbf{u}+\mathbf{v}\) is orthogonal to \(\mathbf{u}-\mathbf{v}\), what can you say about the relative magnitudes of \(\mathbf{u}\) and \(\mathbf{v}\) ?

Step-by-Step Solution

Answer

The magnitudes of \(\mathbf{u}\) and \(\mathbf{v}\) are equal.

Step 1: Understand orthogonality conditions

Two vectors \(\mathbf{a}\) and \(\mathbf{b}\) are orthogonal if their dot product is zero, i.e., \(\mathbf{a} \cdot \mathbf{b} = 0\). In this exercise, we are given that \((\mathbf{u} + \mathbf{v})\) is orthogonal to \((\mathbf{u} - \mathbf{v})\). This implies that \((\mathbf{u} + \mathbf{v}) \cdot (\mathbf{u} - \mathbf{v}) = 0\).

Step 2: Set up the dot product equation

Let's calculate the dot product explicitly: \((\mathbf{u} + \mathbf{v}) \cdot (\mathbf{u} - \mathbf{v}) = (\mathbf{u} \cdot \mathbf{u}) - (\mathbf{u} \cdot \mathbf{v}) + (\mathbf{v} \cdot \mathbf{u}) - (\mathbf{v} \cdot \mathbf{v})\). This can be simplified to \(\mathbf{u} \cdot \mathbf{u} - \mathbf{v} \cdot \mathbf{v}\) because \(\mathbf{u} \cdot \mathbf{v} = \mathbf{v} \cdot \mathbf{u}\).

Step 3: Solve the equation

We then have \(\mathbf{u} \cdot \mathbf{u} - \mathbf{v} \cdot \mathbf{v} = 0\). This implies that \(\mathbf{u} \cdot \mathbf{u} = \mathbf{v} \cdot \mathbf{v}\). Since \(\mathbf{u} \cdot \mathbf{u}\) represents the square of the magnitude of \(\mathbf{u}\) and \(\mathbf{v} \cdot \mathbf{v}\) represents the square of the magnitude of \(\mathbf{v}\), it follows that the magnitudes of \(\mathbf{u}\) and \(\mathbf{v}\) are equal.

Key Concepts

Dot product

The dot product is a fundamental concept in vector algebra, often used to determine the relationship between two vectors. It's an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. For two vectors \(\mathbf{a}\) and \(\mathbf{b}\), the dot product \(\mathbf{a} \cdot \mathbf{b}\) is calculated as:

• Multiplying corresponding components of the vectors.
• Summing up all these products.

Mathematically, it is represented as:\[\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + \ldots + a_nb_n\]When two vectors are orthogonal, their dot product equals zero. This is because the angle between orthogonal vectors is 90 degrees, making the cosine of the angle zero, thus nullifying the dot product. In the context of the given exercise, the orthogonality condition makes \((\mathbf{u} + \mathbf{v}) \cdot (\mathbf{u} - \mathbf{v}) = 0\), which leads us to conclusions about the vector magnitudes.

Vector magnitudes

Magnitude refers to the length or size of a vector and is a crucial aspect of understanding vectors in geometry and physics. The magnitude of a vector \(\mathbf{u}\) is the distance of the vector from the origin point (0) and can be thought of as the 'length' of the vector.To find the magnitude of a vector \(\mathbf{u}\), you calculate the square root of the sum of the squares of its components. It's expressed as:\[||\mathbf{u}|| = \sqrt{u_1^2 + u_2^2 + \ldots + u_n^2}\]In the context of the exercise, the magnitudes of vectors \(\mathbf{u}\) and \(\mathbf{v}\) turn out to be equal. Through mathematical simplification, this is evident from \(\mathbf{u} \cdot \mathbf{u} = \mathbf{v} \cdot \mathbf{v}\). This equation states that the square of the magnitude of \(\mathbf{u}\) equals the square of the magnitude of \(\mathbf{v}\), confirming they have the same size.

Vector operations

Vector operations are the means by which we manipulate vectors to obtain new vectors or scalars. These operations include addition, subtraction, and multiplication (dot and cross products), each with specific rules and results. In this exercise, we are particularly interested in the process of adding and subtracting vectors as it helps us understand vector relationships better. To add two vectors \(\mathbf{u}\) and \(\mathbf{v}\), you simply add their corresponding components:

• If \(\mathbf{u} = [u_1, u_2, \ldots, u_n]\) and \(\mathbf{v} = [v_1, v_2, \ldots, v_n]\), then \(\mathbf{u} + \mathbf{v} = [u_1 + v_1, u_2 + v_2, \ldots, u_n + v_n]\).

Subtraction is similar, where you subtract corresponding components:

• \(\mathbf{u} - \mathbf{v} = [u_1 - v_1, u_2 - v_2, \ldots, u_n - v_n]\).

Utilizing these operations, the exercise shows that \((\mathbf{u} + \mathbf{v})\) is orthogonal to \((\mathbf{u} - \mathbf{v})\). This orthogonality leads us to valuable insights about the vectors' magnitudes and their geometric interpretation.