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Use vectors to find the interior angles of the triangle with the given vertices. (Round your answers to t (-7, -8), (5, 9), (5, 3) (smallest value) ° (largest value)

Use vectors to find the interior angles of the triangle with the given vertices. (Round your answers to t (-7, -8), (5, 9), (5, 3) (smallest value) ° (largest value)

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Finding Interior Angles of a Triangle using Vectors** **Problem Statement:** Use vectors to find the interior angles of the triangle with the given vertices. (Round your answers to two decimal places) Vertices: 1. (-7, -8) 2. (5, 9) 3. (5, 3) **Tasks:** 1. Calculate the smallest interior angle. 2. Calculate the largest interior angle. **Response Fields:** 1. Smallest angle: ____° (smallest value) 2. Second angle: ____° 3. Largest angle: ____° (largest value) To find the interior angles of a triangle using vectors, follow these steps: 1. **Define Vectors:** - Identify the vertices of the triangle as points in a coordinate system. - For the points A(-7, -8), B(5, 9), and C(5, 3), define the vectors connecting these points. 2. **Calculate Vectors:** - Vector AB: From A to B - Vector BC: From B to C - Vector CA: From C to A 3. **Dot Product and Magnitudes:** - Calculate the dot products of each pair of vectors. - Find the magnitudes of the vectors. 4. **Cosine of Angles:** - Use the dot product formula to find the cosines of the angles. - Dot product formula: \( \vec{u} \cdot \vec{v} = |\vec{u}| |\vec{v}| \cos \theta \) 5. **Angles:** - Calculate the angles using the inverse cosine function: \( \theta = \cos^{-1} \left( \frac{\vec{u} \cdot \vec{v}}{|\vec{u}| |\vec{v}|} \right) \) 6. **Convert to Degrees:** - Ensure the angles are in degrees (if using a calculator, ensure it is in degree mode). 7. **Order the Angles:** - Determine which angle is the smallest and largest. Use this method to find the precise values of the angles at each vertex of the triangle formed by the given vertices.
Transcribed Image Text:**Finding Interior Angles of a Triangle using Vectors** **Problem Statement:** Use vectors to find the interior angles of the triangle with the given vertices. (Round your answers to two decimal places) Vertices: 1. (-7, -8) 2. (5, 9) 3. (5, 3) **Tasks:** 1. Calculate the smallest interior angle. 2. Calculate the largest interior angle. **Response Fields:** 1. Smallest angle: ____° (smallest value) 2. Second angle: ____° 3. Largest angle: ____° (largest value) To find the interior angles of a triangle using vectors, follow these steps: 1. **Define Vectors:** - Identify the vertices of the triangle as points in a coordinate system. - For the points A(-7, -8), B(5, 9), and C(5, 3), define the vectors connecting these points. 2. **Calculate Vectors:** - Vector AB: From A to B - Vector BC: From B to C - Vector CA: From C to A 3. **Dot Product and Magnitudes:** - Calculate the dot products of each pair of vectors. - Find the magnitudes of the vectors. 4. **Cosine of Angles:** - Use the dot product formula to find the cosines of the angles. - Dot product formula: \( \vec{u} \cdot \vec{v} = |\vec{u}| |\vec{v}| \cos \theta \) 5. **Angles:** - Calculate the angles using the inverse cosine function: \( \theta = \cos^{-1} \left( \frac{\vec{u} \cdot \vec{v}}{|\vec{u}| |\vec{v}|} \right) \) 6. **Convert to Degrees:** - Ensure the angles are in degrees (if using a calculator, ensure it is in degree mode). 7. **Order the Angles:** - Determine which angle is the smallest and largest. Use this method to find the precise values of the angles at each vertex of the triangle formed by the given vertices.
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