Question

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).

Answer 1

To find the volume of the given parallelepiped, we calculate the scalar triple product of vectors corresponding to the given vertices. The volume equals the absolute value of the scalar triple product, which comes out to be 1 cubic unit.

Step-by-step explanation:

To determine 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), we can use the scalar triple product of vectors.

The vertices given can be thought of as vectors from the origin, specifically A = (1, 4, 0), B = (-2, -5, 2), and C = (-2, 2, 1). To find the volume, we calculate the scalar triple product of these vectors, which is given by (B x C) · A, where 'x' denotes the vector cross product and '·' denotes the dot product.

First, we find the vector cross product of B and C:

• Let B = (-2, -5, 2) and C = (-2, 2, 1)
• The cross product B x C is |i j k|
  | -2 -5 2|
  | -2 2 1|
• Calculating the determinant, we get B x C = ( -5 ∗ 1 - 2 ∗ 2 )i - ( -2 ∗ 1 - 2 ∗ 2 )j + ( -2 ∗ 2 - -5 ∗ -2 )k
• B x C = ( -5 - 4 )i - ( -2 + 4 )j + ( -4 - 10 )k
• B x C = (-9, 2, -14)

Next, we find the dot product of A and (B x C):

• A = (1, 4, 0)
• A · (B x C) = 1 ∗ -9 + 4 ∗ 2 + 0 ∗ -14
• A · (B x C) = -9 + 8 + 0
• A · (B x C) = -1

Therefore, the volume of the parallelepiped is the absolute value of the scalar triple product, which equals |-1| = 1 cubic unit.

Answer 2

17 cubic units.

Step-by-step explanation:

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.