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High School Math Calculus PRECALCULUS W/LIMITS:GRAPH.APPROACH(HS)

To verify: Whether the given points are the vertices of a parallelogram or not.

To verify: Whether the given points are the vertices of a parallelogram or not.

PRECALCULUS W/LIMITS:GRAPH.APPROACH(HS)

PRECALCULUS W/LIMITS:GRAPH.APPROACH(HS)

8th Edition

ISBN: 9781337904285

Author: Larson

Publisher: CENGAGE L

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Question

Chapter 10.3, Problem 48E

(a)

To determine

To verify: Whether the given points are the vertices of a parallelogram or not.

(a)

Expert Solution

Answer to Problem 48E

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

Explanation of Solution

Given information:

The given vertices are A(1,1,1),B(6,5,2),C(7,7,5) and D(2,3,4) .

Formula used: The component form of two point P(x1,y1,z1) , and Q(x2,y2,z2) in space is:

PQ→=〈(x2−x1),(y2−y1),(z2−z1)〉

Calculation:

Calculate the value of component AB→ .

AB→=〈(6−1),(5−1),(2−1)〉=〈5,4,1〉

Calculate the value of component CD→ .

CD→=〈(2−7),(3−7),(4−5)〉=〈−5,−4,−1〉

Calculate the value of component AD→ .

AD→=〈(2−1),(3−1),(4−1)〉=〈1,2,3〉

Calculate the value of component CB→ .

CB→=〈(6−7),(5−7),(2−5)〉=〈−1,−2,−3〉

From the above calculation CD→=−AB→ and CB→=−AD→ . It can be conclude that AB→ is parallel to CD→ and AD→ is parallel to CB→ . So, the given points are vertices of the 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.

(b)

Expert Solution

Answer to Problem 48E

The area of the parallelogram is 283 square units.

Explanation of Solution

Given information:

The vertices of the parallelogramare A(1,1,1),B(6,5,2),C(7,7,5) and D(2,3,4) .

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

Calculation:

The component form of AB→ is 〈5,4,1〉 and it can be written as 5i+4j+k . The component form of AD→ is 〈1,2,3〉 and it can be written as i+2j+3k .

Calculate AB→×AD→ as:

AB→×AD→=|ijk541123|=(12−2)i−(15−1)j+(10−4)k=10i−14j+6k

The area of the parallelogram can be calculated as:

‖AB→×AD→‖=102+142+62=100+196+36=332=283

Therefore, the area of the parallelogram is 283 square units.

Chapter 10.3, Problem 47E

Chapter 10.3, Problem 49E

Chapter 10 Solutions

PRECALCULUS W/LIMITS:GRAPH.APPROACH(HS)

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

Ch. 10.1 - Prob. 11E Ch. 10.1 - Prob. 12E Ch. 10.1 - Prob. 13E Ch. 10.1 - Prob. 14E Ch. 10.1 - Prob. 15E Ch. 10.1 - Prob. 16E Ch. 10.1 - Prob. 17E Ch. 10.1 - Prob. 18E Ch. 10.1 - Prob. 19E Ch. 10.1 - Prob. 20E Ch. 10.1 - Prob. 21E Ch. 10.1 - Prob. 22E Ch. 10.1 - Prob. 23E Ch. 10.1 - Prob. 24E Ch. 10.1 - Prob. 25E Ch. 10.1 - Prob. 26E Ch. 10.1 - Prob. 27E Ch. 10.1 - Prob. 28E Ch. 10.1 - Prob. 29E Ch. 10.1 - Prob. 30E Ch. 10.1 - Prob. 31E Ch. 10.1 - Prob. 32E Ch. 10.1 - Prob. 33E Ch. 10.1 - Prob. 34E Ch. 10.1 - Prob. 35E Ch. 10.1 - Prob. 36E Ch. 10.1 - Prob. 37E Ch. 10.1 - Prob. 38E Ch. 10.1 - Prob. 39E Ch. 10.1 - Prob. 40E Ch. 10.1 - Prob. 41E Ch. 10.1 - Prob. 42E Ch. 10.1 - Prob. 43E Ch. 10.1 - Prob. 44E Ch. 10.1 - Prob. 45E Ch. 10.1 - Prob. 46E Ch. 10.1 - Prob. 47E Ch. 10.1 - Prob. 48E Ch. 10.1 - Prob. 49E Ch. 10.1 - Prob. 50E Ch. 10.1 - Prob. 51E Ch. 10.1 - Prob. 52E Ch. 10.1 - Prob. 53E Ch. 10.1 - Prob. 54E Ch. 10.1 - Prob. 55E Ch. 10.1 - Prob. 56E Ch. 10.1 - Prob. 57E Ch. 10.1 - Prob. 58E Ch. 10.1 - 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