The volume of the parallelepiped having vertices at P (5, 4, 5), Q (4, 10, 6), R (1, 8, 7) and S (2, 6, 9) and edges PQ, PR and PS is

• A: 52 unit
• B: 60 units
• C: 100 units
• D: 108 units

To find the volume of the parallelepiped defined by the points P, Q, R, and S, we first need to determine the vectors corresponding to the edges PQ, PR, and PS.

Step 1: Calculate the Vectors

The vectors can be calculated as follows:

• Vector PQ: Q - P = (4 - 5, 10 - 4, 6 - 5) = (-1, 6, 1)
• Vector PR: R - P = (1 - 5, 8 - 4, 7 - 5) = (-4, 4, 2)
• Vector PS: S - P = (2 - 5, 6 - 4, 9 - 5) = (-3, 2, 4)

Step 2: Set Up the Volume Formula

The volume \( V \) of the parallelepiped can be calculated using the scalar triple product of the vectors:

V = |PQ · (PR × PS)|

Step 3: Calculate the Cross Product

First, we find the cross product of vectors PR and PS:

PR × PS = |i j k|
|-4 4 2|
|-3 2 4|

Calculating this determinant gives:

• i component: (4 * 4 - 2 * 2) = 16 - 4 = 12
• j component: -(-4 * 4 - 2 * -3) = -(-16 + 6) = 10
• k component: (-4 * 2 - 4 * -3) = -8 + 12 = 4

Thus, PR × PS = (12, 10, 4).

Step 4: Calculate the Dot Product

Now, we find the dot product of PQ and (PR × PS):

PQ · (PR × PS) = (-1, 6, 1) · (12, 10, 4) = (-1 * 12) + (6 * 10) + (1 * 4) = -12 + 60 + 4 = 52.

Final Volume Calculation

The volume of the parallelepiped is:

V = |52| = 52 units.

Therefore, the correct answer is A: 52 units.