Find The Volume Of Parallelepiped With Vertices Calculator

Easily find the volume of a parallelepiped defined by four vertices using our calculator.

Calculator

Vertex P Coordinates

Vertex Q Coordinates

Vertex R Coordinates

Vertex S Coordinates

Vectors and Magnitudes

Table showing the components and magnitudes of the vectors formed by the vertices.

Vector	x-component	y-component	z-component	Magnitude
PQ	3.00	1.00	2.00	3.74
PR	1.00	4.00	1.00	4.24
PS	2.00	2.00	5.00	5.74

What is a Volume of Parallelepiped with Vertices Calculator?

A volume of parallelepiped with vertices calculator is a tool used to determine the volume of a parallelepiped when the coordinates of four of its non-coplanar vertices are known. A parallelepiped is a three-dimensional figure formed by six parallelograms, much like a cube is formed by six squares, but the angles are not necessarily 90 degrees. If you know the coordinates of one vertex (say, P) and the three vertices connected to it by edges (Q, R, S), you can define three vectors (PQ, PR, PS) that form the edges of the parallelepiped originating from P. The volume of parallelepiped with vertices calculator uses these coordinates to find the volume.

This calculator is useful for students of geometry, physics, and engineering, as well as professionals who deal with vector mathematics and 3D spaces. It simplifies the process of calculating the volume from vertex coordinates, avoiding manual vector and scalar triple product calculations. Common misconceptions include thinking any four vertices will do (they must define the three edge vectors from one point) or that the order of vertices doesn't matter (it can affect the sign before taking the absolute value).

Volume of Parallelepiped Formula and Mathematical Explanation

To find the volume of a parallelepiped defined by vertices P(Px, Py, Pz), Q(Qx, Qy, Qz), R(Rx, Ry, Rz), and S(Sx, Sy, Sz), we first form three vectors originating from one vertex, say P:

• Vector u = PQ = (Qx – Px, Qy – Py, Qz – Pz) = (u1, u2, u3)
• Vector v = PR = (Rx – Px, Ry – Py, Rz – Pz) = (v1, v2, v3)
• Vector w = PS = (Sx – Px, Sy – Py, Sz – Pz) = (w1, w2, w3)

The volume of the parallelepiped formed by these three vectors is the absolute value of their scalar triple product:

Volume = |u · (v x w)| or |w · (u x v)| or |(u x v) · w|

This scalar triple product can also be calculated as the absolute value of the determinant of the matrix formed by the components of the three vectors:

Volume = | det

Our volume of parallelepiped with vertices calculator performs these vector subtractions and determinant calculation to give you the volume.

Variables Table

Variable	Meaning	Unit	Typical Range
Px, Py, Pz	Coordinates of vertex P	Length units	Real numbers
Qx, Qy, Qz	Coordinates of vertex Q	Length units	Real numbers
Rx, Ry, Rz	Coordinates of vertex R	Length units	Real numbers
Sx, Sy, Sz	Coordinates of vertex S	Length units	Real numbers
u1, u2, u3	Components of vector PQ	Length units	Real numbers
v1, v2, v3	Components of vector PR	Length units	Real numbers
w1, w2, w3	Components of vector PS	Length units	Real numbers
Volume	Volume of the parallelepiped	Cubic length units	Non-negative real numbers

Practical Examples

Let's see how the volume of parallelepiped with vertices calculator works with some examples.

Example 1: A Simple Case

Suppose we have vertices P=(0,0,0), Q=(3,0,0), R=(0,2,0), and S=(0,0,5). The vectors are PQ=(3,0,0), PR=(0,2,0), PS=(0,0,5). Volume = |3(2*5 – 0*0) – 0(0*5 – 0*0) + 0(0*0 – 2*0)| = |3(10)| = 30 cubic units. This is a rectangular parallelepiped (a box) with side lengths 3, 2, and 5.

Example 2: A Slanted Parallelepiped

Consider vertices P=(1,1,1), Q=(4,2,3), R=(2,5,2), and S=(3,3,6). Vector PQ = (3, 1, 2) Vector PR = (1, 4, 1) Vector PS = (2, 2, 5) Volume = |3(4*5 – 1*2) – 1(1*5 – 1*2) + 2(1*2 – 4*2)| = |3(18) – 1(3) + 2(-6)| = |54 – 3 – 12| = |39| = 39 cubic units. Our volume of parallelepiped with vertices calculator would give this result.

How to Use This Volume of Parallelepiped with Vertices Calculator

1. Enter Vertex Coordinates: Input the x, y, and z coordinates for each of the four vertices P, Q, R, and S into the respective fields. Ensure P is the common origin vertex for the three edge vectors PQ, PR, and PS.
2. Calculate: The calculator automatically updates the volume and intermediate values as you type. You can also click the "Calculate Volume" button.
3. View Results: The primary result is the volume, displayed prominently. You'll also see the calculated vectors PQ, PR, PS, and the scalar triple product.
4. Check Table and Chart: The table shows vector components and magnitudes, while the chart visually compares magnitudes and volume.
5. Reset or Copy: Use the "Reset" button to go back to default values or "Copy Results" to copy the data.

Understanding the results helps in visualizing the parallelepiped and its constituent vectors. The volume of parallelepiped with vertices calculator makes this complex calculation straightforward.

Key Factors That Affect Parallelepiped Volume Results

1. Coordinates of Vertices: The primary determinants of the volume are the exact x, y, and z coordinates of the four vertices P, Q, R, and S. Changing any coordinate changes the vectors and thus the volume.
2. Choice of Origin Vertex (P): While the volume remains the same, the intermediate vectors PQ, PR, PS depend on which vertex is chosen as P and which three are Q, R, S relative to P.
3. Relative Positions of Q, R, S to P: The lengths and directions of vectors PQ, PR, and PS dictate the shape and volume.
4. Angle Between Vectors: The angles between the vectors PQ, PR, and PS influence the volume. If the vectors are nearly coplanar, the volume will be small. The volume is maximized when the vectors are mutually orthogonal (for given lengths).
5. Linear Dependence: If the three vectors PQ, PR, and PS are coplanar (linearly dependent), the volume will be zero, meaning the four points lie on the same plane and don't form a 3D parallelepiped.
6. Units Used: The volume will be in "cubic units" corresponding to the units used for the coordinates (e.g., cubic meters if coordinates are in meters).

Using a reliable volume of parallelepiped with vertices calculator ensures accuracy, but understanding these factors is crucial.

Frequently Asked Questions (FAQ)

What if my four vertices are coplanar?

If the four vertices P, Q, R, and S lie on the same plane, the three vectors PQ, PR, and PS will be coplanar, and the scalar triple product (and thus the volume) will be zero. Our volume of parallelepiped with vertices calculator will show a volume of 0.

Does the order of vertices Q, R, S matter?

The order in which you define Q, R, S relative to P will affect the intermediate cross product and the sign of the scalar triple product, but since we take the absolute value for the volume, the final volume will be the same.

What if I have more than four vertices?

A parallelepiped is defined by three edge vectors originating from one vertex. You need to identify one vertex and three adjacent vertices connected to it by edges to use this calculator.

Can I input negative coordinates?

Yes, the coordinates of the vertices can be positive, negative, or zero. The volume of parallelepiped with vertices calculator handles these.

What units will the volume be in?

The volume will be in cubic units based on the units of your input coordinates. If your coordinates are in centimeters, the volume will be in cubic centimeters.

Is this the same as the volume of a rectangular box?

A rectangular box (cuboid) is a special case of a parallelepiped where all angles between adjacent edges are 90 degrees. If the vectors PQ, PR, and PS are mutually orthogonal, the volume will simply be the product of their lengths.

How is the scalar triple product calculated?

It's calculated as the dot product of one vector with the cross product of the other two, or as the determinant of the matrix formed by the three vectors, as explained in the formula section. The volume of parallelepiped with vertices calculator does this automatically.

Can I calculate the surface area with this tool?

No, this calculator is specifically for the volume. Calculating the surface area would require finding the area of the six parallelogram faces, which involves cross products of pairs of vectors.

Related Tools and Internal Resources

• Vector Addition Calculator: Calculate the sum of two or more vectors.
• Dot Product Calculator: Find the dot product of two vectors.
• Cross Product Calculator: Find the cross product of two vectors in 3D space.
• Distance Between Two Points Calculator: Calculate the distance between two points in 2D or 3D.
• Midpoint Calculator: Find the midpoint between two points.
• General Volume Calculator: Calculators for volumes of various shapes.