(a) Explanation Given: | v ⋅ w | ≤ ‖ v ‖ ‖ w ‖ . Proof: The generic formula for a dot product between two vectors v, w ids given as: | v ⋅ w | = ‖ v ‖ ‖ w ‖ cos θ where θ is the angle between the vectors v, w. Recall that the angle θ between any two vectors v and w is given by the relation. cos θ = v ⋅ w ‖ v ‖ ‖ w ‖ (b) To determine To show: The equality | v ⋅ w | = ‖ v ‖ ‖ w ‖ occurs if and only if v = t w for some scalar t . (c) To determine To prove: the triangle inequality ‖ v + w ‖ ≤ ‖ v ‖ + ‖ w ‖ .

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ISBN: 9781465208880

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Chapter 9.3, Problem 58PS

(a)

To determine

To prove: the Cauchy-Schwarz inequality.

(b)

To determine

To show: The equality |v⋅w|=‖v‖‖w‖ occurs if and only if v=tw for some scalar t .

(c)

To determine

To prove: the triangle inequality ‖v+w‖≤‖v‖+‖w‖ .

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Chapter 9.3, Problem 57PS

Chapter 9.3, Problem 59PS

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Chapter 9 Solutions

Calculus

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To prove: the Cauchy-Schwarz inequality.

Solution Summary: The author proves the Cauchy-Schwarz inequality by using the generic formula for a dot product between two vectors v, w.

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To prove: the Cauchy-Schwarz inequality.

To show: The equality |v⋅w|=‖v‖‖w‖ occurs if and only if v=tw for some scalar t .

To prove: the triangle inequality ‖v+w‖≤‖v‖+‖w‖ .

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Chapter 9 Solutions