Cross Product Calculator 3 Vectors?

A cross product calculator for 3 vectors typically computes the scalar triple product, which gives the volume of the parallelepiped formed by three vectors. You enter three vectors (each with x, y, z components) and the calculator returns the scalar triple product, along with the intermediate cross product of the first two vectors. This value tells you the signed volume, and its absolute value is the actual volume.

Formula

The scalar triple product of three vectors A, B, and C is written as A · (B × C). It combines the dot product and the cross product. First, you compute the cross product B × C, which gives a new vector perpendicular to both B and C. Then you take the dot product of that result with A.

In component form, if A = (a_x, a_y, a_z), B = (b_x, b_y, b_z), and C = (c_x, c_y, c_z), the scalar triple product is:

a_x(b_yc_z − b_zc_y) − a_y(b_xc_z − b_zc_x) + a_z(b_xc_y − b_yc_x)

This is equivalent to the determinant of a 3×3 matrix with rows (or columns) being the components of A, B, and C.

The result is a scalar (a single number), not a vector. Its absolute value equals the volume of the parallelepiped (a skewed box) whose three edges are the three vectors. The sign indicates the orientation: positive if the vectors form a right-handed system, negative if left-handed.

Worked Example

Using the default values from the calculator:

• A = (1, 2, 3)
• B = (4, 5, 6)
• C = (7, 8, 9)

First compute B × C:

B × C = (b_yc_z − b_zc_y, b_zc_x − b_xc_z, b_xc_y − b_yc_x)

= (5×9 − 6×8, 6×7 − 4×9, 4×8 − 5×7)

= (45 − 48, 42 − 36, 32 − 35)

= (−3, 6, −3)

Now take the dot product with A:

A · (B × C) = 1×(−3) + 2×6 + 3×(−3) = −3 + 12 − 9 = 0

The scalar triple product is 0. The volume is |0| = 0. This means the three vectors lie in the same plane (they are coplanar) — they do not form a 3D shape.

Try changing C to (7, 8, 10) and you get a non-zero result, meaning the vectors span a volume.

Why the Scalar Triple Product Matters

The scalar triple product appears in physics and engineering whenever you need the volume of a parallelepiped or the condition for coplanarity. For example, in crystallography, the unit cell of a crystal is a parallelepiped defined by three lattice vectors; its volume is the absolute value of the scalar triple product. In fluid mechanics, the scalar triple product shows up in the Jacobian when transforming coordinates. In computer graphics, it is used to determine if three points are collinear or if a set of vertices forms a degenerate polygon.

It also has a geometric interpretation: the cross product B × C gives a vector whose magnitude is the area of the parallelogram spanned by B and C. Multiplying by the component of A perpendicular to that parallelogram gives the height, so the product is base area × height = volume.

How to Use the Calculator

Enter the x, y, z components of each of the three vectors. The calculator accepts any real numbers, including negatives and decimals. The first output is the scalar triple product, which can be negative, zero, or positive. The second output is the absolute value, which is the volume of the parallelepiped.

If you only need the cross product of two vectors (not the triple product), you can use a standard 3D cross product calculator. This page is specifically for the three-vector case, which is a common search query because people often want the volume of a parallelepiped defined by three vectors.

To get the volume of a tetrahedron (a pyramid with a triangular base) formed by three vectors from a common vertex, divide the scalar triple product’s absolute value by 6. That is because a tetrahedron is exactly one-sixth of the parallelepiped.

Common Mistakes and Misconceptions

• Mixing up the order: The scalar triple product is cyclic: A·(B×C) = B·(C×A) = C·(A×B). Swapping any two vectors changes the sign, but the absolute value remains the same. So the volume is unaffected by order, but the sign is not.
• Using the wrong formula: Some people mistakenly compute (A×B)×C, which is a vector triple product, not a scalar. That yields a vector, not a volume.
• Forgetting the absolute value: The scalar triple product can be negative. If you want volume, you must take the absolute value.
• Assuming zero means no volume: A zero scalar triple product means the three vectors are coplanar — they lie in the same plane. The parallelepiped is flat, so its volume is zero. This is a useful check for coplanarity.
• Unit errors: If your vectors represent physical quantities with units (e.g., meters), the scalar triple product will have units of length³ (cubic meters). Be consistent with units across all components.

Relationship to the Cross Product of Two Vectors

For two vectors, the cross product gives a vector perpendicular to both, with magnitude equal to the area of the parallelogram. For three vectors, the scalar triple product is the natural extension: it gives the volume of the parallelepiped. Many students first learn the cross product for two vectors, then encounter the triple product when studying determinants or vector calculus.

If you have three vectors and you want a vector result, you might compute (A×B)×C or A×(B×C). Those are vector triple products, which are different and have their own identities (like the BAC-CAB rule). This calculator focuses on the scalar triple product because that is what the query “cross product calculator 3 vectors” most often refers to.

Limitations of the Formula

The scalar triple product is defined only for vectors in three-dimensional space. In 2D, there is no cross product, and in higher dimensions, the generalization is more complex (using exterior algebra). Also, the formula assumes the vectors are given in Cartesian coordinates with the standard orthonormal basis (i, j, k). If your vectors are in spherical or cylindrical coordinates, you must convert them to Cartesian first.

The calculator gives exact arithmetic results for real numbers, but if you enter very large numbers (beyond about 10^15), floating-point rounding may occur. For most practical purposes, the precision is sufficient.

Frequently Asked Questions

What is the difference between cross product and scalar triple product?

The cross product of two vectors is a vector perpendicular to both. The scalar triple product of three vectors is the dot product of one vector with the cross product of the other two; it is a scalar number representing the signed volume of the parallelepiped.

How do I find the volume of a parallelepiped with three vectors?

Take the absolute value of the scalar triple product: |A · (B × C)|. That gives the volume. If the result is zero, the vectors are coplanar and the volume is zero.

What does a negative scalar triple product mean?

A negative value means the three vectors form a left-handed coordinate system. The absolute value is still the volume. The sign is useful in physics to determine orientation, for example in the right-hand rule.

Can I use this calculator for 2D vectors?

No. The cross product and scalar triple product require three dimensions. For 2D vectors, you can compute the area of a parallelogram using the 2D determinant, but that is a different calculation.

How do I find the volume of a tetrahedron from three vectors?

Divide the absolute value of the scalar triple product by 6. A tetrahedron is one-sixth of the parallelepiped formed by the same three edges.

What if my vectors are not given in Cartesian coordinates?

Convert them to Cartesian coordinates first. For example, spherical coordinates (r, θ, φ) convert to (r sin θ cos φ, r sin θ sin φ, r cos θ). Then apply the scalar triple product formula.

Related

• Angle Between Two Planes Calculator
• Vector Cross Product Calculator (3D)
• 3D Unit Vector Calculator

in ax | A_x | - | 1 | -100000 | 100000
in ay | A_y | - | 2 | -100000 | 100000
in az | A_z | - | 3 | -100000 | 100000
in bx | B_x | - | 4 | -100000 | 100000
in by | B_y | - | 5 | -100000 | 100000
in bz | B_z | - | 6 | -100000 | 100000
in cx | C_x | - | 7 | -100000 | 100000
in cy | C_y | - | 8 | -100000 | 100000
in cz | C_z | - | 9 | -100000 | 100000
out triple | Scalar Triple Product | - | ax*(by*cz - bz*cy) - ay*(bx*cz - bz*cx) + az*(bx*cy - by*cx) | 4
out vol | Volume (absolute) | - | abs(ax*(by*cz - bz*cy) - ay*(bx*cz - bz*cx) + az*(bx*cy - by*cx)) | 4