A rectangular prism is defined by vectors (2,0,0) (0,9,6) and (0,0,3). Find the volume.
To find the volume of a rectangular prism defined by three non-parallel vectors, we can use the triple scalar product or the scalar triple product.
The triple scalar product is the absolute value of the dot product of the first vector with the cross product of the second and third vectors.
Let's calculate it step by step:
1. Find the cross product of the second and third vectors:
v₂ × v₃ = (0, 9, 6) × (0, 0, 3)
To find the cross product of two vectors, we can calculate the determinants:
i j k
0 9 6
0 0 3
= i(9 * 3 - 6 * 0) - j(0 * 3 - 6 * 0) + k(0 * 0 - 9 * 0)
= i(27) - j(0) + k(0)
= (27, 0, 0)
2. Calculate the dot product of the first vector with the cross product:
v₁ ⋅ (v₂ × v₃) = (2, 0, 0) ⋅ (27, 0, 0)
To find the dot product, we multiply the corresponding components and sum them:
(2 * 27) + (0 * 0) + (0 * 0)
= 54 + 0 + 0
= 54
3. Finally, take the absolute value of the dot product to get the volume:
Volume = |v₁ ⋅ (v₂ × v₃)|
= |54|
= 54
Therefore, the volume of the rectangular prism is 54 cubic units.