find the volume of the parallelepiped having u= 3i-5k+k, v=2j-2k and w=3i+j+k as adjacent edges
that would be u•v×w =
|3 -5 1|
|0 2 -2| = 36
|3 1 1|
To find the volume of a parallelepiped given its adjacent edges, you can use the scalar triple product. The scalar triple product of three vectors is defined as follows:
(u.cross(v)).dot(w)
Let's calculate it step by step:
1. Find the cross product of vectors u and v:
u.cross(v) = (3i - 5k + k).cross(2j - 2k)
= (3i - 5k + k).cross(2j - 2k)
= (3i - 5k + k)x(0i + 2j - 2k)
= (3 * (-2)i - (-2)(-5)j + (2 * 2)k)
= (-6i + 10j + 4k + 10j + 4k)
= -6i + 20j + 8k
2. Calculate the dot product of the cross product of u and v with vector w:
(u.cross(v)).dot(w) = (-6i + 20j + 8k).dot(3i + j + k)
= -6 * 3 + 20 * 1 + 8 * 1
= -18 + 20 + 8
= 10
The volume of the parallelepiped is 10 cubic units.