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Chapter 11: Problem 1

Express the dot product of \(\mathbf{u}\) and \(\mathbf{v}\) in terms of their magnitudes and the angle between them.

Short Answer

Expert verified
Answer: The dot product of two vectors, u and v, is equal to the product of their magnitudes and the cosine of the angle between them, expressed as u 路 v = ||u|| * ||v|| * cos(胃).

Step by step solution

01

Recall the Dot Product Definition

The dot product of two vectors \(\mathbf{u}\) and \(\mathbf{v}\), denoted by \(\mathbf{u} \cdot \mathbf{v}\), is defined as the product of the magnitudes of the vectors and the cosine of the angle between them: \(\mathbf{u} \cdot \mathbf{v} = ||\mathbf{u}|| \, ||\mathbf{v}|| \cos{\theta}\) Where \(||\mathbf{u}||\) is the magnitude of vector \(\mathbf{u}\), \(||\mathbf{v}||\) is the magnitude of vector \(\mathbf{v}\), and \(\theta\) is the angle between the two vectors.
02

Solution

To express the dot product of \(\mathbf{u}\) and \(\mathbf{v}\) in terms of their magnitudes and the angle between them, we use the dot product definition from Step 1: \(\mathbf{u} \cdot \mathbf{v} = ||\mathbf{u}|| \, ||\mathbf{v}|| \cos{\theta}\) The dot product of vectors \(\mathbf{u}\) and \(\mathbf{v}\) is equal to the product of their magnitudes and the cosine of the angle between them.

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