Given the points P(-2,3,1), Q(2,-2,0), R(4,1,0). Find the parallelepiped with edge OP, OQ, OR where O is the origin (0,0,0). Volume for a parallelepiped is V=A times B cdot C

Antwan Perez

Antwan Perez

Answered question

2022-10-15

Finding the volume of a parallel piped
Given the points P ( 2 , 3 , 1 ) , Q ( 2 , 2 , 0 ) , R ( 4 , 1 , 0 ).
Find the parallelepiped with edge OP,OQ,OR where O is the origin (0,0,0).
Volume for a parallelepiped is V = A × B C.
So would the volume be O P ( O Q × O R ).
< 2 , 3 , 1 > ( < 2 , 2 , 0 > × < 4 , 1 , 0 > )
< 2 , 3 , 1 > < 0 , 0 , 10 >
area parallelepiped is 10 but would this be correct?

Answer & Explanation

yorbakid2477w6

yorbakid2477w6

Beginner2022-10-16Added 12 answers

Step 1
Looks like you've done it correctly.
The volume of a parallelepiped with three vectors from one vertex P , Q , R is the triple product V = P ( Q × R ):
Step 2
V = ( 2 x ^ + 3 y ^ + z ^ ) ( ( 2 x ^ 2 y ^ ) × ( 4 x ^ + y ^ ) )
V = ( 2 x ^ + 3 y ^ + z ^ ) 10 z ^ = 10.
cousinhaui

cousinhaui

Beginner2022-10-17Added 5 answers

Step 1
Alternately, the volume can be found as (the absolute value of) the determinant of the matrix,
[ 2 3 1 2 2 0 4 1 0 ] ,, which is fairly simple to find via cofactor expansion down the third column.
Step 2
(The determinant approach has the added benefit of being generalizable to n-dimensional parallelpipeds for n 3.)

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