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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.)

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.3: Spanning Sets And Linear Independence
Problem 21EQ
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1. Let v, w be any two vectors. Prove that ||v + w||2 - ||vw||² = 4v • w 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 ||vw|| = ||v + w||. 3. Let x, y be any two vectors. Assume that ||x + y || = |x|| = ||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 axb = a x c, where a × b / 0 and b = c. (This proves that the cross product operation does not have cancellation across equations.)
Transcribed Image Text:1. Let v, w be any two vectors. Prove that ||v + w||2 - ||vw||² = 4v • w 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 ||vw|| = ||v + w||. 3. Let x, y be any two vectors. Assume that ||x + y || = |x|| = ||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 axb = a x 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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