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Let a be the angle between ở and w and ß be the angle between w and ž. Is a > B? Support your answer mathematically.

Let a be the angle between ở and w and ß be the angle between w and ž. Is a > B? Support your answer mathematically.

Trigonometry (11th Edition)
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE: 1. Give the measures of the complement and the supplement of an angle measuring 35°.
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**Problem Statement:** Let \( \alpha \) be the angle between \( \vec{v} \) and \( \vec{w} \), and \( \beta \) be the angle between \( \vec{w} \) and \( \vec{z} \). Is \( \alpha > \beta \)? Support your answer mathematically. --- To determine whether \( \alpha \) is greater than \( \beta \), we can use the concept of dot products in vector mathematics, which relates to the cosine of the angle between two vectors. For any two vectors \( \vec{a} \) and \( \vec{b} \), the dot product is defined as: \[ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos(\theta) \] where \( \theta \) is the angle between \( \vec{a} \) and \( \vec{b} \). 1. **Calculate \( \alpha \):** - Use \( \vec{v} \cdot \vec{w} = |\vec{v}| |\vec{w}| \cos(\alpha) \) 2. **Calculate \( \beta \):** - Use \( \vec{w} \cdot \vec{z} = |\vec{w}| |\vec{z}| \cos(\beta) \) By comparing these cosine values, we can determine the relationship between \( \alpha \) and \( \beta \). - If \(\cos(\alpha) < \cos(\beta)\), then \(\alpha > \beta\). - If \(\cos(\alpha) > \cos(\beta)\), then \(\alpha < \beta\). Based on the given problem, calculate or assume values for the dot products and magnitudes, then analyze the cosine values to conclude if \( \alpha > \beta \).
Transcribed Image Text:**Problem Statement:** Let \( \alpha \) be the angle between \( \vec{v} \) and \( \vec{w} \), and \( \beta \) be the angle between \( \vec{w} \) and \( \vec{z} \). Is \( \alpha > \beta \)? Support your answer mathematically. --- To determine whether \( \alpha \) is greater than \( \beta \), we can use the concept of dot products in vector mathematics, which relates to the cosine of the angle between two vectors. For any two vectors \( \vec{a} \) and \( \vec{b} \), the dot product is defined as: \[ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos(\theta) \] where \( \theta \) is the angle between \( \vec{a} \) and \( \vec{b} \). 1. **Calculate \( \alpha \):** - Use \( \vec{v} \cdot \vec{w} = |\vec{v}| |\vec{w}| \cos(\alpha) \) 2. **Calculate \( \beta \):** - Use \( \vec{w} \cdot \vec{z} = |\vec{w}| |\vec{z}| \cos(\beta) \) By comparing these cosine values, we can determine the relationship between \( \alpha \) and \( \beta \). - If \(\cos(\alpha) < \cos(\beta)\), then \(\alpha > \beta\). - If \(\cos(\alpha) > \cos(\beta)\), then \(\alpha < \beta\). Based on the given problem, calculate or assume values for the dot products and magnitudes, then analyze the cosine values to conclude if \( \alpha > \beta \).
The image displays mathematical expressions representing vectors in two-dimensional space. These vectors are expressed in both component form and in terms of the unit vectors \( \hat{i} \) and \( \hat{j} \). 1. \( \vec{v} = \langle 6, -3 \rangle \) This vector \(\vec{v}\) is written in component form, indicating that it has an x-component of 6 and a y-component of -3. 2. \( \vec{w} = -2\hat{i} + 5\hat{j} \) Vector \(\vec{w}\) is expressed in terms of the unit vectors \(\hat{i}\) and \(\hat{j}\). It has an x-component of -2 and a y-component of 5. 3. \( \vec{z} = -5\hat{j} \) Vector \(\vec{z}\) is fully expressed as \(-5\hat{j}\), indicating that it is oriented entirely along the y-axis with a magnitude of 5 in the negative direction. 4. \( \vec{a} = \langle x, y \rangle \) This vector \(\vec{a}\) is a general representation in component form, where \(x\) and \(y\) are variables representing the components of the vector along the x and y axes, respectively. 5. \( \vec{b} = -y\hat{i} + x\hat{j} \) Vector \(\vec{b}\) is expressed using \(\hat{i}\) and \(\hat{j}\), indicating an x-component of \(-y\) and a y-component of \(x\). There are no graphs or diagrams in this image. The focus is on different ways of expressing vectors mathematically.
Transcribed Image Text:The image displays mathematical expressions representing vectors in two-dimensional space. These vectors are expressed in both component form and in terms of the unit vectors \( \hat{i} \) and \( \hat{j} \). 1. \( \vec{v} = \langle 6, -3 \rangle \) This vector \(\vec{v}\) is written in component form, indicating that it has an x-component of 6 and a y-component of -3. 2. \( \vec{w} = -2\hat{i} + 5\hat{j} \) Vector \(\vec{w}\) is expressed in terms of the unit vectors \(\hat{i}\) and \(\hat{j}\). It has an x-component of -2 and a y-component of 5. 3. \( \vec{z} = -5\hat{j} \) Vector \(\vec{z}\) is fully expressed as \(-5\hat{j}\), indicating that it is oriented entirely along the y-axis with a magnitude of 5 in the negative direction. 4. \( \vec{a} = \langle x, y \rangle \) This vector \(\vec{a}\) is a general representation in component form, where \(x\) and \(y\) are variables representing the components of the vector along the x and y axes, respectively. 5. \( \vec{b} = -y\hat{i} + x\hat{j} \) Vector \(\vec{b}\) is expressed using \(\hat{i}\) and \(\hat{j}\), indicating an x-component of \(-y\) and a y-component of \(x\). There are no graphs or diagrams in this image. The focus is on different ways of expressing vectors mathematically.
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