Question

Find the angle between $\mathbf{v}$ and $\mathrm{w}$ if $\mathbf{v} \cdot \mathrm{w}=-\|\mathbf{v}\|\|\mathrm{w}\|$.

   Find the angle between $\mathbf{v}$ and $\mathrm{w}$ if $\mathbf{v} \cdot \mathrm{w}=-\|\mathbf{v}\|\|\mathrm{w}\|$.

Calculus for AP

Calculus for AP

Jon Rogawski & Ray… 2nd Edition

Chapter 11, Problem 17

Instant Answer

Solved by Expert Bobby Barnes

03/21/2020

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e., $\mathbf{v} \cdot \mathrm{w}=-\|\mathbf{v}\|\|\mathrm{w}\|$. Show more…

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Find the angle between $\mathbf{v}$ and $\mathrm{w}$ if $\mathbf{v} \cdot \mathrm{w}=-\|\mathbf{v}\|\|\mathrm{w}\|$.

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Key Concepts

Relationship between Dot Product and Angle

This relationship shows that the dot product can be used to determine the angle between vectors. If v · w equals the negative product of the magnitudes (i.e., v · w = -||v|| ||w||), it implies that cos? is -1, leading directly to the conclusion that the angle between the vectors is 180°.

Dot Product

The dot product of two vectors is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. It relates to the magnitudes of the vectors and the cosine of the angle between them, encapsulated by the formula v · w = ||v|| ||w|| cos?.

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