Show that if u+ v and u - v are orthogonal, then the two vectors must have the same length. What does this mean with regard to the geometry of the two vectors u and v?

Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Problem 8: Orthogonality and Vector Length**

**Question:**
Show that if \( \mathbf{u} + \mathbf{v} \) and \( \mathbf{u} - \mathbf{v} \) are orthogonal, then the two vectors must have the same length. What does this mean with regard to the geometry of the two vectors \( \mathbf{u} \) and \( \mathbf{v} \)?

**Explanation:**
In this problem, we are asked to demonstrate that if the sum and the difference of two vectors \( \mathbf{u} \) and \( \mathbf{v} \) are orthogonal, then these vectors must have equal magnitudes. Additionally, we are to consider the geometric implications of this property of \( \mathbf{u} \) and \( \mathbf{v} \).

Hints for Solution:
1. **Orthogonality Condition**: Two vectors are orthogonal if their dot product is zero. Hence, \( (\mathbf{u} + \mathbf{v}) \cdot (\mathbf{u} - \mathbf{v}) = 0 \).
2. **Dot Product Expansion**: Expand the dot product \( (\mathbf{u} + \mathbf{v}) \cdot (\mathbf{u} - \mathbf{v}) \) using distributive properties of dot products.
3. **Equality of Magnitudes**: Show that this expansion leads to the conclusion that the magnitudes \( \| \mathbf{u} \| \) and \( \| \mathbf{v} \| \) must be equal.

This problem helps in understanding the relationship between orthogonality and the lengths (magnitudes) of vectors in the context of vector geometry.
Transcribed Image Text:**Problem 8: Orthogonality and Vector Length** **Question:** Show that if \( \mathbf{u} + \mathbf{v} \) and \( \mathbf{u} - \mathbf{v} \) are orthogonal, then the two vectors must have the same length. What does this mean with regard to the geometry of the two vectors \( \mathbf{u} \) and \( \mathbf{v} \)? **Explanation:** In this problem, we are asked to demonstrate that if the sum and the difference of two vectors \( \mathbf{u} \) and \( \mathbf{v} \) are orthogonal, then these vectors must have equal magnitudes. Additionally, we are to consider the geometric implications of this property of \( \mathbf{u} \) and \( \mathbf{v} \). Hints for Solution: 1. **Orthogonality Condition**: Two vectors are orthogonal if their dot product is zero. Hence, \( (\mathbf{u} + \mathbf{v}) \cdot (\mathbf{u} - \mathbf{v}) = 0 \). 2. **Dot Product Expansion**: Expand the dot product \( (\mathbf{u} + \mathbf{v}) \cdot (\mathbf{u} - \mathbf{v}) \) using distributive properties of dot products. 3. **Equality of Magnitudes**: Show that this expansion leads to the conclusion that the magnitudes \( \| \mathbf{u} \| \) and \( \| \mathbf{v} \| \) must be equal. This problem helps in understanding the relationship between orthogonality and the lengths (magnitudes) of vectors in the context of vector geometry.
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