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High School Math Calculus Precalculus with Limits: A Graphing Approach

To check: The triangle formed be the vertices ( 1 , 1 , 3 ) , ( − 4 , 5 , 9 ) and ( 3 , − 1 , 6 ) is an acute triangle, an obtuse triangle or a right triangle.

To check: The triangle formed be the vertices ( 1 , 1 , 3 ) , ( − 4 , 5 , 9 ) and ( 3 , − 1 , 6 ) is an acute triangle, an obtuse triangle or a right triangle.

Precalculus with Limits: A Graphing Approach

Precalculus with Limits: A Graphing Approach

7th Edition

ISBN: 9781305071711

Author: Ron Larson

Publisher: Brooks/Cole

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Question

Chapter 10.2, Problem 64E

To determine

To check: The triangle formed be the vertices (1,1,3) , (−4,5,9) and (3,−1,6) is an acute triangle, an obtuse triangle or a right triangle.

Expert Solution & Answer

Answer to Problem 64E

The triangle formed by joining the vertices (1,1,3) , (−4,5,9) and (3,−1,6) is an obtuse triangle.

Explanation of Solution

Given information:

The vertices of the triangle are (1,1,3) , (−4,5,9) and (3,−1,6) .

Calculation:

Let the points be A(1,1,3) , B(−4,5,9) and C(3,−1,6) be the vertices of the triangle.

The component form of the vector AB is 〈−4−1,5−1,9−3〉=〈−5,4,6〉 .

The component form of the vector BC is 〈3−(−4),−1−5,6−9〉=〈7,−6,−3〉 .

The component form of the vector AC is 〈3−1,−1−1,6−3〉=〈2,−2,3〉 .

Calculate the magnitude of the vector AB .

‖AB‖=(−5)2+(4)2+(6)2=25+16+36=77

Calculate the magnitude of the vector BC .

‖BC‖=(7)2+(−6)2+(−3)2=49+36+9=94

Calculate the magnitude of the vector AC .

‖AC‖=(2)2+(−2)2+(3)2=4+4+9=17

Use the formula for angle between two vectors XY and YZ .

cosθ=XY.YZ‖XY‖‖YZ‖

Calculate the angle between the vectors AB and BC .

cosθ=AB⋅BC‖AB‖‖BC‖=〈−5,4,6〉⋅〈7,−6,−3〉77×94=−5×7+4×−6+6×−377×94=−35−24−1812=−7777×94=−77×9494

The solution for the equation cosθ=−77×9494 is θ=154.82° .

Since one angle is an obtuse and other two angles are acute angles.

Therefore, the triangle formed by joining the vertices (1,1,3) , (−4,5,9) and (3,−1,6) is an obtuse triangle.

Chapter 10.2, Problem 63E

Chapter 10.2, Problem 65E

Chapter 10 Solutions

Precalculus with Limits: A Graphing Approach

Ch. 10.1 - Prob. 1E Ch. 10.1 - Prob. 2E Ch. 10.1 - Prob. 3E Ch. 10.1 - Prob. 4E Ch. 10.1 - Prob. 5E Ch. 10.1 - Prob. 6E Ch. 10.1 - Prob. 7E Ch. 10.1 - Prob. 8E Ch. 10.1 - Prob. 9E Ch. 10.1 - Prob. 10E

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