About the Cross Product Calculator
This cross product calculator multiplies two three-dimensional vectors a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃) and returns the vector a × b, which is perpendicular to both of them. Alongside the result it shows each component worked out from the 3×3 determinant, the magnitude |a × b|, the unit normal vector, the dot product and the angle between the two input vectors.
It is useful for physics and engineering students computing torque (r × F), angular momentum or magnetic force (qv × B), for anyone finding the normal vector of a plane through three points, and for checking linear algebra and multivariable calculus homework. The magnitude also gives the area of the parallelogram spanned by a and b, and half of it is the area of the triangle they form.
The cross product is defined only for 3D vectors (and is anti-commutative: b × a = −(a × b)). For 2D vectors, enter 0 as the third component; the result then points straight along the z-axis and its z-component is the 2D "cross product" a₁b₂ − a₂b₁.
How to use the cross product calculator
- 1Enter the x, y and z components of vector a.
- 2Enter the x, y and z components of vector b (use 0 for z with 2D vectors).
- 3Read a × b and check each component in the working table.
- 4Use the magnitude for the parallelogram area, or half of it for the triangle area.
- 5Use the unit vector when you need a normal direction of length 1.
Formula and method
The cross product is the formal determinant of the 3×3 matrix whose first row is the unit vectors i, j, k, second row is a and third row is b. Expanding along the first row gives the three components above; note the middle (j) term carries a minus sign, which is why it is written as a₃b₁ − a₁b₃.
The result is perpendicular to both a and b, with direction given by the right-hand rule. Its length equals |a||b| sin θ, the area of the parallelogram spanned by the two vectors. The calculator finds the angle θ with atan2(|a × b|, a · b), which stays accurate even when the vectors are almost parallel.
- a, b
- Input vectors (a₁, a₂, a₃) and (b₁, b₂, b₃)
- a × b
- Cross product, a vector perpendicular to a and b
- θ
- Angle between a and b (0° to 180°)
- a · b
- Dot product a₁b₁ + a₂b₂ + a₃b₃
Worked examples
a = (2, 3, 4), b = (5, 6, 7)
x = 3·7 − 4·6 = −3, y = 4·5 − 2·7 = 6, z = 2·6 − 3·5 = −3, so a × b = (−3, 6, −3). Its length is √(9 + 36 + 9) = √54 ≈ 7.3485. With a · b = 56, the vectors are only about 7.48° apart.
Unit vectors i × j
The cross product of the x and y unit vectors is the z unit vector k, which is the right-hand rule in its simplest form. The vectors are perpendicular, so the dot product is 0 and the angle is 90°.
Triangle area from two edge vectors
x = (−1)(−2) − 2·4 = −6, y = 2·1 − 3·(−2) = 8, z = 3·4 − (−1)·1 = 13. |a × b| = √(36 + 64 + 169) = √269 ≈ 16.4012, so the triangle with these two edges has area ≈ 8.2006.
Frequently asked questions
How do you calculate the cross product of two vectors?+
Write the determinant with i, j, k in the first row, a in the second and b in the third, then expand: (a₂b₃ − a₃b₂)i − (a₁b₃ − a₃b₁)j + (a₁b₂ − a₂b₁)k. The result is a vector, not a number.
What is the difference between the dot product and the cross product?+
The dot product gives a scalar equal to |a||b| cos θ and measures how aligned two vectors are. The cross product gives a vector of length |a||b| sin θ that is perpendicular to both inputs, and it is defined for 3D vectors only (a rarely used 7D analogue also exists).
Is the cross product commutative?+
No. It is anti-commutative: b × a = −(a × b). Swapping the order flips the direction of the result while keeping the same length, which is why the right-hand rule depends on order.
What does it mean if the cross product is zero?+
A zero cross product means the vectors are parallel or anti-parallel (or one of them is the zero vector). The angle between them is 0° or 180°, so sin θ = 0 and they span no area.
How do I find the area of a triangle with the cross product?+
Take two edge vectors from the same vertex, compute their cross product and halve its magnitude. For points P, Q, R the area is ½|(Q − P) × (R − P)|.
Can you take the cross product of 2D vectors?+
Not strictly, but you can embed them in 3D with z = 0. The result is (0, 0, a₁b₂ − a₂b₁), and that single number is often called the 2D cross product or perp-dot product.