   Chapter 11.4, Problem 25E

Chapter
Section
Textbook Problem

# Area In Exercises 23 and 24, verify that the points are the vertices of a parallelogram, and find its area. A ( 0 , 3 , 2 ) , B ( 1 , 5 , 5 ) , C ( 6 , 9 , 5 ) , D ( 5 , 7 , 2 )

To determine

To prove: Whether the points A(0,3,2),B(1,5,5),C(6,9,5),D(5,7,2) in the rectangular coordinate system represents the vertices of a parallelogram and find its area.

Explanation

Given:

The vertices of parallelogram are A(0,3,2),B(1,5,5),C(6,9,5),D(5,7,2).

Formula used:

Area of parallelogram:

A=AB×DA

Where AB¯ and DA¯ are non-zero vectors.

Proof:

Calculate four sides of parallelogram correspond to the following vectors as follows:

AB=i+2j+3k

BC=5i+4j+0k

CD=i2j3k

DA=5i4j0k

It implies that AB=CD and BC=DA.

So the vector AB is parallel to the vector CD and similarly the vector BC is parallel to vector DA

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