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Let V be a real vector space with inner product 〈,〉. Prove that for any u,v,w ∈V 〈u,v〉〈u,w〉≤ 1/2(〈v,w〉+ ‖v‖‖w‖ ) ‖u‖^2.

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter5: Inner Product Spaces

Section5.CM: Cumulative Review

Problem 24CM

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Question

Let V be a real vector space with inner product 〈,〉. Prove that for

any u,v,w ∈V

〈u,v〉〈u,w〉≤ 1/2(〈v,w〉+ ‖v‖‖w‖ ) ‖u‖^2.

Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.

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