3D Vector Cross Product Calculator

Find the Cross Product of Two Vectors

3D Vector Cross Product Formula

Formula

\[ \mathbf{C} = \mathbf{A} \times \mathbf{B} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix} \] \[ C_x = A_y B_z - A_z B_y, \quad C_y = A_z B_x - A_x B_z, \quad C_z = A_x B_y - A_y B_x \] \[ |\mathbf{C}| = \sqrt{C_x^2 + C_y^2 + C_z^2} \]

Where:

\(A_x, A_y, A_z\) — components of the first vector \(\mathbf{A}\)
\(B_x, B_y, B_z\) — components of the second vector \(\mathbf{B}\)
\(C_x, C_y, C_z\) — components of the cross product \(\mathbf{C}\)
\(|\mathbf{C}|\) — magnitude (length) of the cross product vector

The cross product \(\mathbf{A} \times \mathbf{B}\) produces a new vector that is perpendicular to both \(\mathbf{A}\) and \(\mathbf{B}\), with a magnitude equal to the area of the parallelogram they span.

3D Vector Cross Product – Calculation Example

Example: find the cross product of \(\mathbf{A} = (2, 3, 4)\) and \(\mathbf{B} = (5, 6, 7)\).

\[ C_x = A_y B_z - A_z B_y = 3 \cdot 7 - 4 \cdot 6 = 21 - 24 = -3 \]

\[ C_y = A_z B_x - A_x B_z = 4 \cdot 5 - 2 \cdot 7 = 20 - 14 = 6 \]

\[ C_z = A_x B_y - A_y B_x = 2 \cdot 6 - 3 \cdot 5 = 12 - 15 = -3 \]

\[ |\mathbf{C}| = \sqrt{(-3)^2 + 6^2 + (-3)^2} = \sqrt{54} \approx 7.348 \]

Result: \(\mathbf{A} \times \mathbf{B} = (-3,\ 6,\ -3)\), with magnitude \(\approx 7.348\).

How to Use the 3D Vector Cross Product Calculator

Enter the three components of the first vector in the fields A — X component, A — Y component and A — Z component.
Enter the three components of the second vector in the fields B — X component, B — Y component and B — Z component.
Negative values and zeros are allowed — just type the minus sign where needed.
Read the result under Your Cross Product: the X, Y and Z components of the resulting vector, its magnitude, and a step-by-step breakdown of every calculation.

What Is the Cross Product?

The cross product (also called the vector product) is an operation on two vectors in three-dimensional space. Unlike the dot product, which returns a single number, the cross product returns a vector. Its direction is perpendicular to the plane containing both input vectors, determined by the right-hand rule: point the fingers of your right hand along \(\mathbf{A}\), curl them toward \(\mathbf{B}\), and your thumb points along \(\mathbf{A} \times \mathbf{B}\).

The order of the vectors matters. Swapping them reverses the sign of every component: \(\mathbf{B} \times \mathbf{A} = -(\mathbf{A} \times \mathbf{B})\). This property, called anticommutativity, means the cross product is not commutative — entering the vectors in a different order gives a different answer.

The magnitude of the cross product has a geometric meaning:

\[ |\mathbf{A} \times \mathbf{B}| = |\mathbf{A}|\,|\mathbf{B}| \sin\theta \]

where \(\theta\) is the angle between the two vectors. This equals the area of the parallelogram formed by \(\mathbf{A}\) and \(\mathbf{B}\). If the result is the zero vector, the two vectors are parallel (or one of them is zero), because \(\sin 0° = 0\). The larger the angle between the vectors, the larger the cross product — it reaches its maximum when they are perpendicular.

Cross products appear throughout physics and engineering: torque is \(\mathbf{r} \times \mathbf{F}\), magnetic force involves \(\mathbf{v} \times \mathbf{B}\), and angular momentum is \(\mathbf{r} \times \mathbf{p}\). In computer graphics, cross products are used to compute surface normals for lighting and shading.

Assumptions, Limitations and Practical Tips

Three dimensions only. The cross product is defined only for 3D vectors (a limited version exists for 2D vectors, returning a scalar). This calculator requires all three components of each vector.
Order matters. If you get the opposite sign from a textbook answer, check whether the book computed \(\mathbf{B} \times \mathbf{A}\) instead of \(\mathbf{A} \times \mathbf{B}\).
Perpendicularity check. A quick way to verify the result: the dot products \(\mathbf{A} \cdot \mathbf{C}\) and \(\mathbf{B} \cdot \mathbf{C}\) should both equal zero, since the cross product is perpendicular to both inputs.
Parallel vectors. If the output is the zero vector \((0, 0, 0)\), the inputs are parallel or one of them is zero — this is a valid result, not an error.
Units. The components should use consistent units. If \(\mathbf{A}\) is in meters and \(\mathbf{B}\) is in newtons, the cross product is in newton-meters (N·m), the unit of torque.

If you need to brush up on the basic arithmetic behind the component multiplications, the multiplication calculator and the subtraction calculator can handle the individual steps.

Frequently Asked Questions

How do I calculate a cross product by hand without a calculator?

Use the cofactor expansion along the top row of the 3×3 determinant: multiply and subtract the pairs of products for each component. A helpful memory aid for the middle component: its sign is negative in the determinant expansion, which is why \(C_y = A_z B_x - A_x B_z\) has the terms in an order opposite to the other two. Write each product out explicitly before subtracting — most hand-calculation errors are simple sign and order slips.

What is the difference between the cross product and the dot product?

The dot product takes two vectors and returns a scalar (a single number), computed as \(\mathbf{A} \cdot \mathbf{B} = |A||B|\cos\theta\). It measures how much the vectors point in the same direction. The cross product returns a vector perpendicular to both inputs, with magnitude \(|A||B|\sin\theta\). Dot product: alignment. Cross product: perpendicularity and rotation effects like torque.

Why is my answer the negative of the textbook's answer?

Almost always because the order of the vectors was swapped. The cross product is anticommutative: \(\mathbf{B} \times \mathbf{A} = -(\mathbf{A} \times \mathbf{B})\), so every component flips sign. Check which vector your source lists first. If the order matches, re-check the middle component \(C_y\) — its formula is the one most often written with the terms in the wrong order.

Can I use this calculator for 2D vectors?

Yes — set the Z components of both vectors to 0. The result will always point along the Z axis: for \(\mathbf{A} = (A_x, A_y, 0)\) and \(\mathbf{B} = (B_x, B_y, 0)\), the cross product is \((0, 0, A_x B_y - A_y B_x)\). This scalar Z value is exactly the 2D "scalar cross product" used in geometry, and its absolute value is the area of the parallelogram spanned by the two vectors.

How do I find the angle between two vectors using the cross product?

Compute \(\sin\theta = |\mathbf{A} \times \mathbf{B}| / (|\mathbf{A}|\,|\mathbf{B}|)\), where the numerator is the magnitude shown by this calculator and \(|\mathbf{A}|\), \(|\mathbf{B}|\) are the lengths of the inputs. Then \(\theta = \arcsin(\sin\theta)\). Note the limitation: the cross product alone cannot distinguish \(\theta\) from \(180° - \theta\). For a unique angle between 0° and 180°, combine it with the dot product or use the atan2 method.

How can I get the area of a triangle from the cross product?

The magnitude \(|\mathbf{A} \times \mathbf{B}|\) equals the area of the parallelogram spanned by the two vectors. A triangle is exactly half of that parallelogram, so the triangle area is \(\frac{1}{2}|\mathbf{A} \times \mathbf{B}|\). For a triangle with vertices \(P\), \(Q\), \(R\), take \(\mathbf{A} = Q - P\) and \(\mathbf{B} = R - P\), then halve the magnitude from this calculator.

Does the calculator work with decimals and fractions?

Yes, all input fields accept decimal numbers, including negative values and zeros — type the minus sign directly. The results are shown with up to three decimal places. If you need exact fractions, convert them to decimals first (e.g., \(\frac{1}{3} \approx 0.333\)) — but be aware that rounding the inputs slightly changes the exact result, so keep enough decimal places for your required precision.