# Determine the vectors representing the diagonals and the length of the diagonals of the following: A parallelogram is formed in R^3 by the vectors PA = (3,2,-3) and PB=(4, 1, 5).  The point P = (0, 2, 3). We have PA = (3,2,-3) and PB =( 4,1,5) and P = (0,2,3)

The vertexes of the parallelogram are:

• A = (0 , 2 , 3) + ( 3 , 2 , -3) = (3 , 4 ,0)
• B =(0 , 2, 3) + (4, 1, 5) = (4, 3,...

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We have PA = (3,2,-3) and PB =( 4,1,5) and P = (0,2,3)

The vertexes of the parallelogram are:

• A = (0 , 2 , 3) + ( 3 , 2 , -3) = (3 , 4 ,0)
• B =(0 , 2, 3) + (4, 1, 5) = (4, 3, 8)
• The third vertex is P = ( 0, 2, 3)
• The fourth vertex C = A + PB = (0, 2, 3) + (3, 2, -3)= (7, 5, 5)

The diagonals of the parallelogram are:

• AB = (3 , 4 ,0) - (4, 3, 8) = ( -1, 1, -8).

The length of AB is sqrt [ 1 + 1 + 64] = sqrt 66

• The other diagonal CP = (7, 5, 5) - ( 0, 2, 3) = ( 7, 3, 2)

The length of CP is sqrt [ 49 + 9 + 4] = sqrt 62.

The required diagonals and their lengths are:

• ( -1, 1, -8), length sqrt 66
• ( 7, 3, 2), length sqrt 62.
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