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 10.3, Problem 47E

(a)

To determine

To verify:Whether the given points are the vertices of a parallelogram.

(a)

Expert Solution
Check Mark

Answer to Problem 47E

It is verified that the given points are the vertices of a parallelogram.

Explanation of Solution

Given information:

The vertices are A(2,−1,4),B(3,1,2),C(0,5,6) , and D(−1,3,8) .

Calculation:

If there are two points in space P(x1,y1,z1) , and Q(x2,y2,z2) .

The component form is,

  PQ→=〈(x2−x1),(y2−y1),(z2−z1)〉

Calculate the value of component AB→ .

  AB→=〈(3−2),(1+1),(2−4)〉=(1,2,−2)

Calculate the value of component CD→ .

  CD→=〈(−1−0),(3−5),(8−6)〉=(−1,−2,2)

Calculate the value of component AD→ .

  AD→=〈(−1−2),(3+1),(8−4)〉=(−3,4,4)

Calculate the value of component CB→ .

  CB→=〈(3−0),(1−5),(2−6)〉=(3,−4,−4)

From the above calculation CD→=−AB→ and CB→=−AD→ it conclude that AB→ is parallel to CD→ and AD→ is parallel to CB→ . Hence, the given points are vertices of the a parallelogram with AB→ and AD→ as adjacent sides.

Therefore, it is verified that the given points are the vertices of a parallelogram.

(b)

To determine

To find: The area of the parallelogram with the given adjacent sides.

(b)

Expert Solution
Check Mark

Answer to Problem 47E

The area of the parallelogram with adjacent sides A(2,−1,4),B(3,1,2),C(0,5,6) and D(−1,3,8) is 610 square unit .

Explanation of Solution

Given information:

The adjacent sides of the parallelogram are vectors A(2,−1,4),B(3,1,2),C(0,5,6) and D(−1,3,8) .

Formula used:Let u and v be the adjacent sides of a parallelogram. The area of a parallelogram is ‖u×v‖ .

Calculation:

It is known that AB and CD are the adjacent sides of the parallelogram.

Calculate AB→×AD→ as:

  AB→×AD→=|ijk12−2−344|=(8+8)i−(4−6)j+(4+6)k=16i+2j+10k ki8

The area of the parallelogram can be calculated as:

  ‖AB×AD‖=162+22+102=256+4+100=360=610

Therefore, the area of the parallelogram with adjacent sides A(2,−1,4),B(3,1,2),C(0,5,6) and D(−1,3,8) is 610 square unit .

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Chapter 10.3, Problem 46E
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Chapter 10.3, Problem 48E

Chapter 10 Solutions

Precalculus with Limits: A Graphing Approach

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