Find the volume of the parallelepiped with the given vertices:
A(0,0,0) B(4,0,0) C(4,-2,3) D(0,-2,3)
E(4,5,3) F(0,5,3) G(0,3,6) H(4,3,6)
in ABCD, let
u = B-A = (4,0,0)
v = D-A = (0,-2,3)
Now using the other parallel face,
w = E-A = (4,5,3)
volume is u×v•w
Thank you so very much!
To find the volume of a parallelepiped, we need to find the area of the base and multiply it by the height. In this case, the base is a parallelogram given by vectors AB and AD, and the height is the distance between the base and the opposite face.
Step 1: Find the vectors AB and AD.
Vector AB: (4-0, 0-0, 0-0) = (4, 0, 0)
Vector AD: (0-0, -2-0, 3-0) = (0, -2, 3)
Step 2: Find the cross product of vectors AB and AD.
Cross product: AB x AD = (0 * 3 - (-2) * 0, (4 * 3 - 0 * 0), (4 * (-2) - 0 * (-2))) = (0, 12, -8)
Step 3: Find the magnitude of the cross product.
Magnitude: |AB x AD| = √(0^2 + 12^2 + (-8)^2) = √(144 + 64) = √208
Step 4: Find the area of the base.
Area of the base: |AB x AD| = √208
Step 5: Find the height.
The height is the distance between the base and the opposite face. Let's find the distance between the points G(0,3,6) and E(4,5,3).
Step 5.1: Find the vector GE.
Vector GE = (4-0, 5-3, 3-6) = (4, 2, -3)
Step 5.2: Find the magnitude of vector GE.
Magnitude: |GE| = √(4^2 + 2^2 + (-3)^2) = √(16 + 4 + 9) = √29
Step 6: Calculate the volume.
Volume = Area of the base * Height
Volume = √208 * √29
Volume = √(208 * 29)
Volume = √6032
Volume ≈ 77.71 (rounded to two decimal places)