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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Chapter 10.2, Problem 63E
To determine

To check: The triangle formed be the vertices (2,−3,4) , (0,1,2) and (−1,2,0) is an acute triangle, an obtuse triangle or a right triangle.

Expert Solution & Answer
Check Mark

Answer to Problem 63E

The triangle formed by joining the vertices (2,−3,4) , (0,1,2) and (−1,2,0) is an obtuse triangle.

Explanation of Solution

Given information:

The vertices of the triangle are (2,−3,4) , (0,1,2) and (−1,2,0) .

Calculation:

Let the points be A(2,−3,4) , B(0,1,2) and C(−1,2,0) be the vertices of the triangle.

The component form of the vector AB is 〈0−2,1−(−3),2−4〉=〈−2,4,−2〉 .

The component form of the vector BC is 〈−1−0,2−1,0−2〉=〈−1,1,−2〉 .

The component form of the vector AC is 〈−1−2,2−(−3),0−4〉=〈−3,5,−4〉 .

Calculate the magnitude of the vector AB .

  ‖AB‖=(−2)2+(4)2+(−2)2=4+16+4=24=26

Calculate the magnitude of the vector BC .

  ‖BC‖=(−1)2+(1)2+(−2)2=1+1+4=6

Calculate the magnitude of the vector AC .

  ‖AC‖=(−3)2+(5)2+(−4)2=9+25+16=50=52

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‖=〈−2,4,−2〉⋅〈−1,1,−2〉26×6=−2×−1+4×1+(−2)×−212=2+4+412=1012=56

The solution for the equation cosθ=56 is θ=33.56° . So, the angle between the vectors AB and BC is 33.56° .

Calculate the angle between the vectors AB and AC .

  cosθ=AB⋅AC‖AB‖‖AC‖=〈−2,4,−2〉⋅〈−3,5,−4〉26×52=−2×−3+4×5+(−2)×−4203=6+20+8203=34360

The solution for the equation cosθ=34360 is θ=11.04° . So, the angle between the vectors AB and AC is 11.04° .

The sum of all the internal angles of triangle is 180° . So, the third angle of triangle is,

  θ=180°−33.56°−11.04°=135.4° .

Since one angle is an obtuse angle and other two angles are acute angle. So, the formed triangle is an obtuse angle.

Therefore, the triangle formed by joining the vertices (2,−3,4) , (0,1,2) and (−1,2,0) is an obtuse triangle.

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Chapter 10 Solutions

Precalculus with Limits: A Graphing Approach

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