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Determine whether or not the points A(1,3,−2), B(3,4,1), C(2,0,−2), and D(4,8,4) are coplanar. If not, find the volume of the pyramid with these four points as its vertices, given that its volume is one-sixth that of the parallelepiped spanned by AB, AC, and AD

Question
Determine whether or not the points A(1,3,−2), B(3,4,1), C(2,0,−2), and D(4,8,4) are coplanar. If not, find the volume of the pyramid with these four points as its vertices, given that its volume is one-sixth that of the parallelepiped spanned by AB, AC, and AD
 
check_circleAnswer
Step 1

Let vector a , b and c represents the vector vector AB,AC and AD respectively.

a-3i+4j+k-(i+3j-2) i+j+3k
5-2i +0j-2k-(i+3j-2k)-i-3j+0k
C 4i+8j+4k-(i+3j-2k)-3i +5j+6k
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Image Transcriptionclose

a-3i+4j+k-(i+3j-2) i+j+3k 5-2i +0j-2k-(i+3j-2k)-i-3j+0k C 4i+8j+4k-(i+3j-2k)-3i +5j+6k

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Step 2

Now we find scalar triple product of vectors a, b and c

12
1
3
a (бiхд)-
3 5
-з 0-2(-18-0)- (6-0) +3(5+9)
а.(bx
1
6
a (бхд)-0
help_outline

Image Transcriptionclose

12 1 3 a (бiхд)- 3 5 -з 0-2(-18-0)- (6-0) +3(5+9) а.(bx 1 6 a (бхд)-0

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Step 3

Since scalar triple product of vectors a, b and c is 0, ...

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