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?
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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F333de6cc-3da3-4bcc-9995-6ee1769198c2%2F45d058bb-c033-42c1-8fc3-ae9ce76dc80b%2F1g2l8yb.jpeg&w=3840&q=75)
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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