1. Let v, w be any two vectors. Prove that ||v + w||² - ||v − w||² = 4v. w W (Hint: Use properties of the dot product. Do NOT use component forms of the vectors.)
1. Let v, w be any two vectors. Prove that ||v + w||² - ||v − w||² = 4v. w W (Hint: Use properties of the dot product. Do NOT use component forms of the vectors.)
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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![1. Let **v, w** be any two vectors. Prove that
\[
\| \mathbf{v} + \mathbf{w} \|^2 - \| \mathbf{v} - \mathbf{w} \|^2 = 4 \mathbf{v} \cdot \mathbf{w}
\]
*(Hint: Use properties of the dot product. Do NOT use component forms of the vectors.)*
2. Suppose that **v, w** are orthogonal. Use Equation (1) to show that
\[
\| \mathbf{v} - \mathbf{w} \| = \| \mathbf{v} + \mathbf{w} \|
\]
3. Let **x, y** be any two vectors. Assume that
\[
\| \mathbf{x} + \mathbf{y} \| = \| \mathbf{x} \| = \| \mathbf{y} \|
\]
Find the angle between **x, y**.
*(Hint: Begin by squaring both sides of the first equation given above. Alternatively, use geometry as a guide. Your proof cannot rely on geometry, but it is a strong starting point.)*
4. Use the 3-space standard basis vectors to find three nonzero vectors **a, b, c** satisfying **a × b = a × c**, where **a × b ≠ 0** and **b ≠ c**.
*(This proves that the cross product operation does not have cancellation across equations.)*](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F464fe16a-00d5-4182-8379-61bd4b8bbe02%2F6a1c12b2-4f69-4c93-9c79-3b0976c8332a%2Fd5x6zsh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. Let **v, w** be any two vectors. Prove that
\[
\| \mathbf{v} + \mathbf{w} \|^2 - \| \mathbf{v} - \mathbf{w} \|^2 = 4 \mathbf{v} \cdot \mathbf{w}
\]
*(Hint: Use properties of the dot product. Do NOT use component forms of the vectors.)*
2. Suppose that **v, w** are orthogonal. Use Equation (1) to show that
\[
\| \mathbf{v} - \mathbf{w} \| = \| \mathbf{v} + \mathbf{w} \|
\]
3. Let **x, y** be any two vectors. Assume that
\[
\| \mathbf{x} + \mathbf{y} \| = \| \mathbf{x} \| = \| \mathbf{y} \|
\]
Find the angle between **x, y**.
*(Hint: Begin by squaring both sides of the first equation given above. Alternatively, use geometry as a guide. Your proof cannot rely on geometry, but it is a strong starting point.)*
4. Use the 3-space standard basis vectors to find three nonzero vectors **a, b, c** satisfying **a × b = a × c**, where **a × b ≠ 0** and **b ≠ c**.
*(This proves that the cross product operation does not have cancellation across equations.)*
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