Find The Volume Of The Parallelepiped Defined By The Vectors

Finding the Volume of a Parallelepiped Defined by Vectors

The volume of a parallelepiped is a fundamental concept in linear algebra and vector calculus with applications spanning various fields, including physics and engineering. Understanding how to calculate this volume, given three vectors defining its edges, is crucial for many problem-solving scenarios. This comprehensive guide will delve into the methods for calculating the volume, explore the underlying mathematical principles, and provide practical examples to solidify your understanding.

Understanding Parallelepipeds and Vectors

A parallelepiped is a three-dimensional figure formed by six parallelograms. Think of it as a skewed box. It's defined by three vectors, a, b, and c, which represent the edges emanating from a single vertex. These vectors are not necessarily orthogonal (perpendicular) to each other.

Vectors, in this context, are mathematical objects possessing both magnitude (length) and direction. They are often represented as ordered triplets (a₁, a₂, a₃), (b₁, b₂, b₃), and (c₁, c₂, c₃) in three-dimensional space.

The Scalar Triple Product: The Key to Volume Calculation

The volume of the parallelepiped defined by vectors a, b, and c is directly related to the scalar triple product. The scalar triple product is a scalar (a single number) obtained by taking the dot product of one vector with the cross product of the other two. Mathematically, it's represented as:

V = a ⋅ (b × c)

Where:

• V represents the volume of the parallelepiped.
• a ⋅ (b × c) denotes the scalar triple product.
• × denotes the cross product of two vectors.
• ⋅ denotes the dot product of two vectors.

Let's break down the individual operations:

The Cross Product (b × c)

The cross product of two vectors, b and c, results in a new vector that is perpendicular to both b and c. Its magnitude represents the area of the parallelogram formed by b and c. The cross product is calculated as:

b × c = (b₂c₃ - b₃c₂)i + (b₃c₁ - b₁c₃)j + (b₁c₂ - b₂c₁)k

Where:

• i, j, and k are the unit vectors along the x, y, and z axes, respectively.

The Dot Product (a ⋅ (b × c))

The dot product of vector a and the resulting vector from the cross product (b × c) is a scalar value. It represents the projection of vector a onto the vector (b × c). The dot product is calculated as:

**a ⋅ (b × c) = a₁ (b₂c₃ - b₃c₂) + a₂ (b₃c₁ - b₁c₃) + a₃ (b₁c₂ - b₂c₁) **

Calculating the Volume: A Step-by-Step Approach

To find the volume of the parallelepiped, follow these steps:

1. Determine the vectors: Identify the three vectors, a, b, and c, that define the edges of the parallelepiped.

2. Calculate the cross product: Compute the cross product of vectors b and c using the formula mentioned above.

3. Calculate the dot product: Compute the dot product of vector a with the result of the cross product (b × c).

4. Determine the volume: The absolute value of the scalar triple product is the volume of the parallelepiped. The absolute value is taken because volume cannot be negative. Therefore:

   V = |a ⋅ (b × c)|

Practical Examples

Let's illustrate the process with a few examples:

Example 1:

Let a = (1, 2, 3), b = (4, 5, 6), and c = (7, 8, 9).

1. Cross product (b × c): (59 - 68)i + (67 - 49)j + (48 - 57)k = -3i + 6j - 3k = (-3, 6, -3)

2. Dot product (a ⋅ (b × c)): (1)(-3) + (2)(6) + (3)(-3) = -3 + 12 - 9 = 0

3. Volume: |0| = 0. This indicates that the three vectors are coplanar (lie on the same plane), resulting in a parallelepiped with zero volume (essentially a flat parallelogram).

Example 2:

Let a = (1, 0, 0), b = (0, 2, 0), and c = (0, 0, 3).

1. Cross product (b × c): (03 - 00)i + (00 - 03)j + (00 - 20)k = (0, 0, 0)

2. Dot product (a ⋅ (b × c)): (1)(0) + (0)(0) + (0)(0) = 0

3. Volume: |0| = 0. This is because the vectors are coplanar in this instance.

Example 3:

Let a = (1, 1, 1), b = (2, 0, 1), and c = (1, 2, 0).

1. Cross product (b × c): (00 - 12)i + (11 - 20)j + (22 - 01)k = (-2, 1, 4)

2. Dot product (a ⋅ (b × c)): (1)(-2) + (1)(1) + (1)(4) = 3

3. Volume: |3| = 3 cubic units.

Geometrical Interpretation and Applications

The scalar triple product has a strong geometrical interpretation. Its absolute value represents the volume of the parallelepiped formed by the three vectors. If the scalar triple product is zero, it means the three vectors are coplanar, and the parallelepiped collapses into a two-dimensional shape, possessing no volume.

This concept finds widespread applications in various fields:

• Physics: Calculating the volume of a crystal lattice, determining the torque of a force acting on a rigid body.

• Engineering: Computing volumes of irregularly shaped objects, analyzing stress and strain in structures.

• Computer Graphics: Calculating the volume of 3D models, performing transformations on 3D objects.

• Linear Algebra: Determining linear independence of vectors, calculating determinants of 3x3 matrices.

Determinant Method: An Alternative Approach

The scalar triple product is directly related to the determinant of a 3x3 matrix formed by the components of the vectors:

V = |det(A)|

Where A is the matrix:

| a₁  a₂  a₃ |
| b₁  b₂  b₃ |
| c₁  c₂  c₃ |

Calculating the determinant provides an alternative way to find the volume of the parallelepiped. The determinant is calculated as:

det(A) = a₁(b₂c₃ - b₃c₂) - a₂(b₁c₃ - b₃c₁) + a₃(b₁c₂ - b₂c₁)

Notice that this is exactly the same as the scalar triple product formula.

Conclusion

Calculating the volume of a parallelepiped defined by three vectors is a fundamental concept with significant practical implications. By understanding the scalar triple product or utilizing the determinant method, you can effectively determine the volume, even in scenarios involving non-orthogonal vectors. Remember, the absolute value of the result always represents the volume, ensuring a positive value that reflects the physical reality of the three-dimensional space occupied by the parallelepiped. The understanding of this concept is pivotal for advanced studies in linear algebra, vector calculus, and their applications in various scientific and engineering disciplines. Mastering this calculation enhances problem-solving skills and provides a solid foundation for more complex mathematical concepts.