   Chapter 11.4, Problem 26E

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 ( 2 , − 3 , 1 ) , B ( 6 , 5 , − 1 ) , C ( 7 , 2 , 2 ) , D ( 3 , − 6 , 4 )

To determine

To prove: Whether the points A(2,3,1),B(6,5,1),C(7,2,2),D(3,6,4) in the rectangular coordinate system represents the vertices of parallelogram and find its area.

Explanation

Given:

The four vertices of parallelogram are A(2,3,1),B(6,5,1),C(7,2,2),D(3,6,4).

Formula used:

Area of parallelogram:

A=AB×DA

Proof:

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

AB=4i+8j2k

BC=i3j+3k

CD=4i8j+2k

DA=i+3j3k

From the above, 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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