Explanation Approach: A vector from the initial point A ( x 1 , x 2 ) and the terminal point B ( y 1 , y 2 ) is A B → = 〈 y 1 − x 1 , y 2 − x 2 〉 . The dot product of two vectors, u = 〈 u 1 , u 2 〉 = u 1 i + u 2 j and v = 〈 v 1 , v 2 〉 = v 1 i + v 2 j , is defined as u ⋅ v = u 1 v 1 + u 2 v 2 . The length or magnitude of a vector u is denoted as ‖ u ‖ and is defined as ‖ u ‖ = u ⋅ u . The magnitude of a vector is always positive. The angle θ between two non-zero vectors u and v can be found by using the formula, cos θ = u ⋅ v ‖ u ‖ ‖ v ‖ . Given Information: The vertices of a triangle are A ( − 3 , − 4 ) , B (1, 7) and C (8, 2). Calculation: A vector from A ( − 3 , − 4 ) to B (1, 7) is A B = 〈 1 − ( − 3 ) , 7 − ( − 4 ) 〉 = 〈 4 , 11 〉 . A vector from B (1, 7) to C (8, 2) is B C = 〈 8 − 1 , 2 − 7 〉 = 〈 7 , − 5 〉 . A vector from A ( − 3 , − 4 ) to C (8, 2) is A C = 〈 8 − ( − 3 ) , 2 − ( − 4 ) 〉 = 〈 11 , 6 〉 . A B ⋅ A C = 〈 4 , 11 〉 ⋅ 〈 11 , 6 〉 = ( 4 ) ( 11 ) + ( 11 ) ( 6 ) = 44 + 66 = 110 A B ⋅ B C = 〈 4 , 11 〉 ⋅ 〈 7 , − 5 〉 = ( 4 ) ( 7 ) + ( 11 ) ( − 5 ) = 28 + ( − 55 ) = − 27 ‖ A B ‖ 2 = A B ⋅ A B = 〈 4 , 11 〉 ⋅ 〈 4 , 11 〉 = ( 4 ) ( 4 ) + ( 11 ) ( 11 ) = 16 + 121 = 137 Therefore, ‖ A B ‖ = 137 . ‖ B C ‖ 2 = B C ⋅ B C = 〈 7 , − 5 〉 ⋅ 〈 7 , − 5 〉 = ( 7 ) ( 7 ) + ( − 5 ) ( − 5 ) = 49 + 25 = 74 Therefore, ‖ B C ‖ = 74

10th Edition

ISBN: 9781337271073

Author: Ron Larson

Publisher: Cengage Learning

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Question

Chapter 6.4, Problem 44E

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To find:

The interior angles of the triangle whose vertices are (−3,−4), (1, 7) and (8, 2) using the concept of vector.

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Chapter 6.4, Problem 43E

Chapter 6.4, Problem 45E

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Chapter 6 Solutions

Precalculus (MindTap Course List)

Ch. 6.1 - Checkpoint For the triangle shown, A=30, B=45, and...Ch. 6.1 - Prob. 2CP Ch. 6.1 - Prob. 3CP Ch. 6.1 - Prob. 4CP Ch. 6.1 - Prob. 5CP Ch. 6.1 - Prob. 6CP Ch. 6.1 - Prob. 7CP Ch. 6.1 - Prob. 1E Ch. 6.1 - Prob. 2E Ch. 6.1 - Prob. 3E

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The interior angles of the triangle whose vertices are ( − 3 , − 4 ) , (1, 7) and (8, 2) using the concept of vector .

Solution Summary: The author explains how to find the interior angles of the triangle whose vertices are (-3,-4), (1, 7) and (8, 2) using the concept of vector.

To find:

The interior angles of the triangle whose vertices are (−3,−4), (1, 7) and (8, 2) using the concept of vector.

Want to see the full answer?

Chapter 6 Solutions

The interior angles of the triangle whose vertices are ( − 3 , − 4 ) , (1, 7) and (8, 2) using the concept of vector .

Solution Summary: The author explains how to find the interior angles of the triangle whose vertices are (-3,-4), (1, 7) and (8, 2) using the concept of vector.

To find: The interior angles of the triangle whose vertices are (−3,−4), (1, 7) and (8, 2) using the concept of vector.

Want to see the full answer?

Chapter 6 Solutions

To find:

The interior angles of the triangle whose vertices are (−3,−4), (1, 7) and (8, 2) using the concept of vector.