What this scalar triple product calculator does

This calculator takes three three-dimensional vectors and computes their scalar triple product, written as A · (B × C). From this single number, you get the signed volume of the parallelepiped defined by the three vectors and information about their orientation (right-handed or left-handed). The absolute value of the result is the volume; the sign tells you how the vectors are arranged in 3D space.

To use the tool, enter each vector as three comma-separated numbers, such as 1,2,3. The calculator then parses these components, forms the appropriate determinant, and outputs both the scalar triple product and the corresponding volume.

Definition of the scalar triple product

Given three vectors in 3D,

$\vec{a} = (a_{1}, a_{2}, a_{3})$
$\vec{b} = (b_{1}, b_{2}, b_{3})$
$\vec{c} = (c_{1}, c_{2}, c_{3})$

their scalar triple product is defined as

$\vec{a} \cdot (\vec{b} \times \vec{c}) .$

This is a single real number. It is called “scalar” because the result is a scalar, not a vector, and “triple product” because it involves three vectors combined through a cross product and a dot product.

Geometric meaning: from vectors to volume

The scalar triple product connects vector algebra with 3D geometry. Geometrically:

The cross product $\vec{b} \times \vec{c}$ is a vector perpendicular to the plane spanned by $\vec{b}$ and $\vec{c}$ . Its magnitude equals the area of the parallelogram built on $\vec{b}$ and $\vec{c}$ .
The dot product $\vec{a} \cdot (\vec{b} \times \vec{c})$ projects $\vec{a}$ onto this perpendicular direction. The magnitude of this projection is the height of the parallelepiped formed by $\vec{a}, \vec{b}, \vec{c}$ .

Combining these observations:

Base area = $∥ \vec{b} \times \vec{c} ∥$
Height = the component of $\vec{a}$ along $\vec{b} \times \vec{c}$
Volume = base area × height = $| a \cdot (b \times c) |$

The sign of the scalar triple product tells you orientation:

If $\vec{a}, \vec{b}, \vec{c}$ follow the right-hand rule, the scalar triple product is positive.
If they follow a left-handed orientation, the scalar triple product is negative.
If the scalar triple product is zero, the three vectors are coplanar (they lie in the same plane), and the “parallelepiped” has zero volume.

Determinant formula

The scalar triple product can be written as the determinant of a 3×3 matrix. A concise way to express this is:

$\vec{a} \cdot (\vec{b} \times \vec{c}) = det (\begin{matrix} a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3} \end{matrix}) .$

In expanding this determinant, you obtain an explicit formula in terms of the components:

$\vec{a} \cdot (\vec{b} \times \vec{c}) = a_{1} (b_{2} c_{3} - b_{3} c_{2}) - a_{2} (b_{1} c_{3} - b_{3} c_{1}) + a_{3} (b_{1} c_{2} - b_{2} c_{1})$

This expansion shows several key properties:

The result is linear in each vector: doubling $\vec{a}$ doubles the scalar triple product, and similarly for $\vec{b}$ and $\vec{c}$ .
Swapping any two rows (i.e., swapping any two vectors) changes the sign of the determinant and therefore the sign of the scalar triple product.
If the rows are linearly dependent (one row is a linear combination of the others), the determinant is zero. This corresponds to the vectors being coplanar in geometry.

MathML representation of the formula

For completeness and better support in some mathematical browsers and tools, here is the scalar triple product expressed using MathML. This block encodes the determinant form:

V = | \begin{matrix} a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3} \end{matrix} |

Here V denotes the scalar triple product value, which equals the signed volume of the parallelepiped formed by $\vec{a}, \vec{b}, \vec{c}$ .

How this calculator works (conceptual steps)

Behind the scenes, the calculator follows these steps:

Parse inputs. Each vector field is split on commas to extract the three components. The tool expects exactly three numeric values for each vector.
Convert to numbers. The components are converted to real numbers. Negative values and decimals are allowed as long as they follow standard numeric notation.
Form the determinant. With $a_{1}, a_{2}, a_{3}$ , $b_{1}, b_{2}, b_{3}$ , and $c_{1}, c_{2}, c_{3}$ in hand, the algorithm computes the determinant using the expanded formula above.
Compute the scalar triple product. The determinant value is the scalar triple product $\vec{a} \cdot (\vec{b} \times \vec{c})$ .
Compute the volume. The volume of the parallelepiped is the absolute value of this result, $| V |$ .
Check special cases. If the value is exactly zero (or extremely close to zero within numerical tolerance), the vectors are treated as coplanar and the volume as zero.

The implementation details may vary, but conceptually the determinant formula provides a direct and efficient way to obtain the scalar triple product from the vector components.

How to interpret the results

Once you run the calculator, you will typically see at least two key numbers and possibly some explanatory text:

Scalar triple product: This is the raw value of $\vec{a} \cdot (\vec{b} \times \vec{c})$ . It may be positive, negative, or zero.
Volume: The calculator usually reports the volume as the absolute value of the scalar triple product, $| V |$ , which is always non-negative.
Orientation or coplanarity message (if provided): Based on the sign or the magnitude of the result, the tool may indicate whether the vectors are right-handed, left-handed, or coplanar.

In practical terms:

If $V > 0$ , the triple $(\vec{a}, \vec{b}, \vec{c})$ is right-handed, and the volume is $V$ in magnitude.
If $V < 0$ , the triple is left-handed, and the volume is $- V$ in magnitude.
If $V = 0$ , the vectors are coplanar and do not span a 3D volume; the parallelepiped collapses to a flat shape.

Worked example

Consider the vectors

$\vec{a} = (1, 0, 0)$
$\vec{b} = (0, 2, 0)$
$\vec{c} = (0, 0, 3)$

These vectors are aligned with the coordinate axes and have lengths 1, 2, and 3, respectively.

Compute the cross product $\vec{b} \times \vec{c}$ .
$\vec{b} \times \vec{c} = (2 \cdot 3 - 0 \cdot 0, 0 \cdot 0 - 0 \cdot 3, 0 \cdot 0 - 2 \cdot 0) = (6, 0, 0) .$
Dot with $\vec{a}$ .
$\vec{a} \cdot (\vec{b} \times \vec{c}) = (1, 0, 0) \cdot (6, 0, 0) = 1 \cdot 6 + 0 \cdot 0 + 0 \cdot 0 = 6 .$
Interpretation.
The scalar triple product is 6, so the volume of the parallelepiped is $| 6 | = 6$ in whatever cubic units the original vectors use. The positive sign indicates a right-handed orientation.

If you enter these vectors into the calculator as

1,0,0
0,2,0
0,0,3

you should obtain a scalar triple product of 6 and a volume of 6.

Comparison with related vector operations

The scalar triple product is closely related to more familiar operations like the dot product and cross product. The following table highlights the differences and connections.

Operation	Inputs	Output type	Main geometric meaning	Typical use
Dot product	2 vectors	Scalar	Measures how aligned two vectors are; related to the cosine of the angle between them.	Checking orthogonality, projecting one vector onto another, work in physics.
Cross product	2 vectors (in 3D)	Vector	Perpendicular to both inputs; magnitude equals area of the parallelogram formed by the vectors.	Finding normal vectors, torque, rotational effects, 3D graphics.
Scalar triple product	3 vectors (in 3D)	Scalar	Signed volume of the parallelepiped spanned by the three vectors.	Computing volumes, checking coplanarity, determining orientation in 3D.

Practical uses of the scalar triple product

The scalar triple product appears in many areas of science and engineering:

Geometry and analytic geometry: Quickly compute the volume of a parallelepiped or determine whether three vectors in space are coplanar.
Physics: Evaluate quantities involving torque and moment arms, or compute oriented volumes when analyzing fluxes or triple integrals with Jacobians.
Computer graphics and 3D modeling: Determine the orientation of triangles and faces, detect back-facing polygons, and check whether sets of edges form degenerate (flat) shapes.
Engineering and robotics: Analyze the configuration of arms or supports in 3D, assess whether structures span full 3D space, and detect singular configurations where motion becomes constrained.

In many of these contexts, an automated calculator helps avoid sign errors and arithmetic mistakes, especially when working under time pressure or with non-integer coordinates.

Assumptions and limitations of this calculator

For reliable results, keep in mind the following assumptions and limitations:

Dimension: The calculator is designed for 3D vectors only. Each input vector must have exactly three components.
Input format: Enter components as comma-separated numbers, for example 1,2,3 or -0.5, 2.75, 4. Extra commas or missing components will typically produce an error or invalid result.
Numeric types: Real numbers (including decimals and negatives) are allowed. Very large or very small magnitudes may be affected by standard floating-point rounding and precision limits.
Units: The calculator does not track units. If your vector components carry units (for example, meters), the resulting volume will inherit the corresponding cubic units (for example, cubic meters), but this interpretation is up to you.
Exact coplanarity: For theoretical work, vectors are coplanar when the scalar triple product is exactly zero. With floating-point arithmetic, you may see very small nonzero values (like 1e-15) instead of perfect zero.
Error handling: Non-numeric entries, incomplete vectors, or formatting errors may prevent the calculator from producing a meaningful result. Always double-check your input if the output seems unexpected.

Tips for using the calculator effectively

Double-check that each vector has three components before pressing the compute button.
If you are using the result in a physical context, remember to assign and track your own units.
For nearly coplanar vectors, expect small numerical noise; treat sufficiently tiny results as effectively zero, depending on your tolerance.
When comparing orientations, pay attention to the sign of the scalar triple product, not just its magnitude.

Related concepts

The scalar triple product fits naturally into a broader toolkit of vector operations. If you are studying this topic, you may also want to explore dot products, cross products, vector norms, and matrix determinants, as they all interact in the geometry of Euclidean 3-space.

Scalar Triple Product Calculator

What this scalar triple product calculator does

Definition of the scalar triple product

Geometric meaning: from vectors to volume

Determinant formula

MathML representation of the formula

How this calculator works (conceptual steps)

How to interpret the results

Worked example

Comparison with related vector operations

Practical uses of the scalar triple product

Assumptions and limitations of this calculator

Tips for using the calculator effectively

Related concepts

Embed this calculator

Scalar Triple Product Calculator

What this scalar triple product calculator does

Definition of the scalar triple product

Geometric meaning: from vectors to volume

Determinant formula

MathML representation of the formula

How this calculator works (conceptual steps)

How to interpret the results

Worked example

Comparison with related vector operations

Practical uses of the scalar triple product

Assumptions and limitations of this calculator

Tips for using the calculator effectively

Related concepts

Embed this calculator

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