Volume of Parallelepiped with Vectors Calculator
Easily calculate the volume of a parallelepiped defined by three 3D vectors using the scalar triple product with our Volume of Parallelepiped with Vectors Calculator.
| Vector | x (i) | y (j) | z (k) |
|---|---|---|---|
| u | 1 | 0 | 0 |
| v | 0 | 1 | 0 |
| w | 0 | 0 | 1 |
| v × w | 0 | 0 | 1 |
Magnitudes of vectors u, v, w, and v × w.
What is a Volume of Parallelepiped with Vectors Calculator?
A Volume of Parallelepiped with Vectors Calculator is a tool used to determine the volume of a parallelepiped (a three-dimensional figure formed by six parallelograms) when it is defined by three vectors originating from the same point, representing its adjacent edges. The calculator uses the concept of the scalar triple product of these three vectors to find the volume. The volume is the absolute value of the scalar triple product.
This calculator is useful for students, engineers, physicists, and anyone working with 3D geometry and vector algebra. It simplifies the calculation, which can otherwise be done by finding the determinant of a 3×3 matrix formed by the vector components.
Common misconceptions include thinking the order of vectors matters for the volume (it only affects the sign of the scalar triple product, but the volume is the absolute value) or that the vectors must be orthogonal (they don't have to be).
Volume of Parallelepiped with Vectors Formula and Mathematical Explanation
The volume (V) of a parallelepiped formed by three vectors u = (u1, u2, u3), v = (v1, v2, v3), and w = (w1, w2, w3) is given by the absolute value of their scalar triple product:
V = |u ⋅ (v × w)|
The cross product v × w is calculated first:
v × w = (v2*w3 – v3*w2)i + (v3*w1 – v1*w3)j + (v1*w2 – v2*w1)k
Then, the dot product of u with (v × w) is calculated:
u ⋅ (v × w) = u1*(v2*w3 – v3*w2) + u2*(v3*w1 – v1*w3) + u3*(v1*w2 – v2*w1)
This scalar triple product is also equal to the determinant of the matrix whose rows (or columns) are the components of the vectors u, v, and w:
det | u1 u2 u3 | | v1 v2 v3 | | w1 w2 w3 | = u1(v2w3 – v3w2) – u2(v1w3 – v3w1) + u3(v1w2 – v2w1)
The volume is the absolute value of this determinant. Our Volume of Parallelepiped with Vectors Calculator performs these steps.
Variables Table
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| u1, u2, u3 | Components of vector u | (length unit) | Any real number |
| v1, v2, v3 | Components of vector v | (length unit) | Any real number |
| w1, w2, w3 | Components of vector w | (length unit) | Any real number |
| V | Volume of the parallelepiped | (length unit)³ | Non-negative real number |
Practical Examples (Real-World Use Cases)
Using the Volume of Parallelepiped with Vectors Calculator is straightforward.
Example 1: Orthogonal Vectors
Suppose we have three vectors representing the edges of a rectangular box (a special parallelepiped):
- u = (3, 0, 0)
- v = (0, 4, 0)
- w = (0, 0, 5)
Inputting these into the calculator:
v × w = (0*5 – 0*0, 0*0 – 3*5, 3*0 – 0*0) – wait, it's v x w: (4*5 – 0*0, 0*0 – 0*5, 0*0 – 4*0) = (20, 0, 0)
u ⋅ (v × w) = 3*20 + 0*0 + 0*0 = 60
Volume = |60| = 60 cubic units. This matches the simple length × width × height for a box.
Example 2: Non-Orthogonal Vectors
Consider vectors:
- u = (1, 2, 0)
- v = (1, 1, 1)
- w = (0, 3, 1)
Using the Volume of Parallelepiped with Vectors Calculator:
v × w = (1*1 – 1*3, 1*0 – 1*1, 1*3 – 1*0) = (-2, -1, 3)
u ⋅ (v × w) = 1*(-2) + 2*(-1) + 0*3 = -2 – 2 + 0 = -4
Volume = |-4| = 4 cubic units.
How to Use This Volume of Parallelepiped with Vectors Calculator
- Enter Vector Components: Input the x (i), y (j), and z (k) components for each of the three vectors u, v, and w into their respective fields.
- Calculate: Click the "Calculate Volume" button or observe the real-time update as you enter values.
- View Results: The calculator will display:
- The primary result: The Volume of the parallelepiped.
- Intermediate results: The components of the cross product v × w and the value of the scalar triple product u ⋅ (v × w).
- A table summarizing the input vectors and v × w.
- A chart showing vector magnitudes.
- Interpret: The volume represents the space enclosed by the parallelepiped defined by the three vectors.
- Reset: Click "Reset" to clear the fields and start a new calculation with default values.
This Volume of Parallelepiped with Vectors Calculator gives you a quick and accurate result.
Key Factors That Affect Volume of Parallelepiped Results
The volume calculated by the Volume of Parallelepiped with Vectors Calculator depends directly on the components of the three input vectors. Here are key factors:
- Magnitude of the Vectors: Longer vectors generally lead to a larger volume, assuming the angles between them don't shrink the enclosed space too much.
- Angles Between the Vectors: The volume is maximized when the three vectors are mutually orthogonal (like the edges of a box). As the angles between them decrease, or if they become more coplanar, the volume decreases. If the three vectors are coplanar, the volume is zero.
- Components of u: The x, y, and z components of vector u directly influence the scalar triple product.
- Components of v: The components of vector v affect the cross product v × w, and thus the volume.
- Components of w: Similarly, the components of vector w are crucial for the v × w cross product and the final volume.
- Linear Dependence: If the three vectors are linearly dependent (one can be expressed as a linear combination of the other two, meaning they lie in the same plane), the volume will be zero. The Volume of Parallelepiped with Vectors Calculator will show this.
Frequently Asked Questions (FAQ)
- What is a parallelepiped?
- A parallelepiped is a three-dimensional figure formed by six parallelograms, where opposite faces are parallel and equal in size and shape.
- What is the scalar triple product?
- The scalar triple product of three vectors u, v, and w is given by u ⋅ (v × w). Geometrically, its absolute value represents the volume of the parallelepiped formed by the three vectors. Our Volume of Parallelepiped with Vectors Calculator uses this.
- Does the order of vectors matter in the scalar triple product for volume?
- The order affects the sign of the scalar triple product (e.g., u ⋅ (v × w) = -v ⋅ (u × w)), but since the volume is the absolute value, the volume remains the same regardless of the order as long as it's a cyclic permutation (u,v,w; v,w,u; w,u,v give same sign, others give opposite).
- What if the volume is zero?
- A volume of zero means the three vectors are coplanar (lie in the same plane), and they do not form a 3D parallelepiped with non-zero volume.
- Can the components of the vectors be negative?
- Yes, vector components can be positive, negative, or zero, representing direction along the axes.
- What units is the volume in?
- The volume will be in cubic units of whatever length unit was used for the vector components (e.g., cubic meters if components were in meters). The Volume of Parallelepiped with Vectors Calculator doesn't assume units, so be consistent.
- How is this related to the determinant of a matrix?
- The scalar triple product u ⋅ (v × w) is equal to the determinant of the 3×3 matrix formed by the components of u, v, and w as its rows (or columns).
- Can I use this calculator for a cube or rectangular box?
- Yes, a cube and a rectangular box are special cases of a parallelepiped where the edges are mutually orthogonal. Just input the vectors along the x, y, and z axes with lengths equal to the sides.
Related Tools and Internal Resources
- Scalar Triple Product Calculator: Directly calculate the scalar triple product u · (v × w).
- Vector Cross Product Calculator: Calculate the cross product of two vectors.
- 3D Vector Calculator: Perform various operations on 3D vectors like addition, subtraction, dot product, and cross product.
- Determinant Calculator: Find the determinant of a 3×3 matrix, useful for volume calculation.
- Geometric Volume Calculator: Calculate volumes of various standard geometric shapes.
- Vector Algebra Tool: Learn more about the basics of vector algebra and its applications.