Find the volume of the parallelepiped with one vertex at the origin and adjacent vertices at (1, 4, 0), (-2, -5, 2), and (-2, 2, 1).

17 cubic units.

The volume (V) of a parallelepiped with one vertex at the origin is given by the absolute value of the scalar vector product of the vectors at the adjacent vertices.

V = |(a x b) · c|

In this case,

a = (1,4,0) = i + 4k + 0j

b = (-2,-5,2) = -2i -5j + 2k

c = (-2,2,1) = -2i + 2j + k

First, let's calculate the cross product of vectors a and b. i.e a x b as follows:

(i) Arrange the vectors in a matrix form

a x b = | i        j         k |

          | 1       4        0 |

          | -2    -5        2 |

(ii) Calculate the determinant of the matrix

a x b = i(8 - 0) -j(2-0) + k(-5+8)

a x b = i(8) -j(2) + k(3)

a x b = 8i - 2j + 3k

Secondly, calculate the scalar product of the cross product found above and the vector c as follows;

(a x b) . c = (8i - 2j + 3k) · (-2i + 2j + k)

Multiply like terms

(a x b) . c = (8 * -2)(i.i) + (-2 * 2)(j.j) + (3 * 1)(k.k)             [i.i = j.j = k.k = 1]

(a x b) . c = (-16) + (-4) + (3)

(a x b) . c = -16 - 4 + 3

(a x b) . c = -17

Thirdly, find the absolute value of the result found above. i.e

|(a x b) . c| = |-17|

|(a x b) . c| = 17

Therefore, the volume of the parallelepiped is 17 cubic units.