Problem 52 - Essential Calculus Early Transcendentals

Question

Show that if \(\mathbf{u}+\mathbf{v}\) and \(\mathbf{u}-\mathbf{v}\) are orthogonal, then the vectors \(\mathbf{u}\) and \(\mathbf{v}\) must have the same length.

Step-by-Step Solution

Answer

If \(\mathbf{u} + \mathbf{v}\) and \(\mathbf{u} - \mathbf{v}\) are orthogonal, then \(\|\mathbf{u}\| = \|\mathbf{v}\|\).

Step 1: Understand Orthogonality

Two vectors are orthogonal if their dot product is zero. This means \((\mathbf{u} + \mathbf{v}) \cdot (\mathbf{u} - \mathbf{v}) = 0\).

Step 2: Expand the Dot Product

Apply the distributive property to expand the dot product: \((\mathbf{u} + \mathbf{v}) \cdot (\mathbf{u} - \mathbf{v}) = \mathbf{u} \cdot \mathbf{u} - \mathbf{u} \cdot \mathbf{v} + \mathbf{v} \cdot \mathbf{u} - \mathbf{v} \cdot \mathbf{v}\).

Step 3: Simplify Using Properties of Dot Products

Since \(\mathbf{u} \cdot \mathbf{v} = \mathbf{v} \cdot \mathbf{u}\), simplify the expression to: \(\mathbf{u} \cdot \mathbf{u} - \mathbf{v} \cdot \mathbf{v} = 0\). This implies that \(\|\mathbf{u}\|^2 - \|\mathbf{v}\|^2 = 0\).

Step 4: Conclude Lengths of Vectors

The equation \(\|\mathbf{u}\|^2 - \|\mathbf{v}\|^2 = 0\) implies that \(\|\mathbf{u}\|^2 = \|\mathbf{v}\|^2\). Therefore, the lengths of \(\mathbf{u}\) and \(\mathbf{v}\) are equal, i.e., \(\|\mathbf{u}\| = \|\mathbf{v}\|\).

Key Concepts

Dot Product

In vector mathematics, the dot product is an essential operation that combines two vectors to produce a scalar, which can be very informative. If you have two vectors, say \( \mathbf{a} \) and \( \mathbf{b} \), the dot product is calculated as \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + \ldots + a_nb_n \) for corresponding components of the two vectors, where \( n \) is the dimension of the vectors.
This operation gives you three important possibilities:
- If the dot product is zero, the vectors are orthogonal, meaning they are at a right angle to one another.
- If the dot product is positive, it indicates that the vectors point somewhat in the same direction.
- If the dot product is negative, it shows the vectors point somewhat in opposite directions.
Orthogonality plays a pivotal role in problems involving vector simplification. This principle is used to establish other relationships, such as vector length equality.

Vector Length Equality

Vector length, or magnitude, is a measure of how much of the vector space a vector spans, which can help you understand how large or small a vector is within its n-dimensional space. The magnitude of a vector \( \mathbf{u} \) is given by \( \|\mathbf{u}\| = \sqrt{u_1^2 + u_2^2 + \ldots + u_n^2} \).
For two vectors to have equal lengths means \( \|\mathbf{u}\| = \|\mathbf{v}\| \). This concept can be shown through the orthogonality condition described in the exercise:
- If \( (\mathbf{u} + \mathbf{v}) \) and \( (\mathbf{u} - \mathbf{v}) \) are orthogonal, their dot product is zero: \( (\mathbf{u} + \mathbf{v}) \cdot (\mathbf{u} - \mathbf{v}) = 0 \).
- Expanding this, using the distributive property of dot products, and simplifying using \( \mathbf{u} \cdot \mathbf{v} = \mathbf{v} \cdot \mathbf{u} \), leads to the realization that the lengths must be equal.
Therefore, you end up with \( \|\mathbf{u}\|^2 = \|\mathbf{v}\|^2 \), confirming they have equal magnitudes.

Properties of Dot Products

The properties of dot products serve as essential tools for simplifying expressions and establishing relationships between multiple vectors. Some of these properties are:
- **Commutative Property:** \( \mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a} \). This property supports various rearrangements in expressions.
- **Distributive Property:** \( \mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = \mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c} \). It is useful when expanding dot products of sums or differences.
- **Orthogonality Condition:** If \( \mathbf{a} \cdot \mathbf{b} = 0 \), the vectors are orthogonal.
- **Magnitude:** The dot product of a vector with itself gives the square of its magnitude: \( \mathbf{a} \cdot \mathbf{a} = \|\mathbf{a}\|^2 \).
These properties enable us to draw conclusions about the relationships and configurations of vectors in space. They are crucial in simplifying vector expressions, especially in proving vector length equality through orthogonal conditions.