2D and 3D Vector Calculator

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cross product calculator

Cross product, parallelogram area and triple product

Switch the dimension to 3D, type A and B (and C for the triple product) and pick the Cross product drawing to see A × B leaving perpendicular to the plane of the two, with the camera you rotate.

The determinant behind the perpendicular vector

In 3D the cross product is A × B = (a₂b₃ - a₃b₂, a₃b₁ - a₁b₃, a₁b₂ - a₂b₁), the symbolic determinant with i, j, k on the first row. The result is a vector perpendicular to both A and B, pointing by the right-hand rule: the fingers sweep from A to B and the thumb points along A × B. Swapping the order flips the sign, A × B = -(B × A), and the product of a vector with itself or with a multiple of itself is the zero vector. A sanity check: i × j = k, that is, (1, 0, 0) × (0, 1, 0) = (0, 0, 1).
The magnitude |A × B| = |A||B| sin θ is the area of the parallelogram built on A and B, and half of it is the triangle area. For A = (3, 4, 2) and B = (-1, 2, 3) the cross product is (8, -11, 10) and the parallelogram area is √(64 + 121 + 100) = √285 ≈ 16.8819. In 2D only the z component exists, a₁b₂ - a₂b₁, and its absolute value is the area: for (2, 1) and (1, 3) it gives 5, and the triangle has 5/2. This is the same determinant used for triangle area from coordinates in the analytic geometry calculator.
A third vector brings in the triple product [A, B, C] = A · (B × C), the 3 by 3 determinant of the components. The absolute value is the parallelepiped volume and one sixth of it is the tetrahedron volume. For A = (3, 4, 2), B = (-1, 2, 3) and C = (1, 0, 2): B × C = (4, 5, -2) and A · (B × C) = 12 + 20 - 4 = 28, so the parallelepiped has volume 28 and the tetrahedron 14/3. The sign depends on the order and the orientation of the basis. If the triple product is zero the three vectors are coplanar, and the calculator flags that case with a notice.

Checked examples

Standard basis

Input: A = (1, 0, 0), B = (0, 1, 0)
Expected output: A × B = (0, 0, 1)

i × j = k.

Radical in the area

Input: A = (3, 4, 2), B = (-1, 2, 3)
Expected output: A × B = (8, -11, 10); área = √285

Triangle: √285/2.

Perpendicular axes

Input: A = (2, 0, 0), B = (0, 3, 0)
Expected output: A × B = (0, 0, 6); área = 6

2 by 3 rectangle.

2D case

Input: A = (2, 1), B = (1, 3)
Expected output: A × B = 5; triângulo = 5/2

Only the z component.

General vectors

Input: A = (1, 2, 3), B = (4, -5, 6)
Expected output: A × B = (27, 6, -13); área = √934

≈ 30.5614.

Triple product

Input: A = (3, 4, 2), B = (-1, 2, 3), C = (1, 0, 2)
Expected output: [A, B, C] = 28; V = 28; tetraedro = 14/3

Positively oriented basis.

Full tool FAQ

How do you add and subtract vectors?

Add or subtract component by component: A + B = (a₁ + b₁, a₂ + b₂) and, in 3D, also a₃ + b₃. For A = (3, 4) and B = (-1, 2): A + B = (2, 6) and A - B = (4, 2). Geometrically, the sum is the diagonal of the parallelogram built on A and B, or the vector that closes the chain when you place the tail of B on the tip of A.

How do you calculate the magnitude and the unit vector?

How do you calculate the dot product and the angle between two vectors?

How do you calculate the cross product and the parallelogram area?

What is the triple product and what is it for?

How do you project one vector onto another?

How do you convert magnitude and angle into components?

How do you find the resultant of several forces and the equilibrant force?

How can you tell whether two vectors are parallel, orthogonal or coplanar?

Is the result exact? What are the limits?

Frequently asked questions

Why does the cross product only exist in 3D?

Because only in 3D (and, in a special way, in 7D) is there a single direction perpendicular to two given ones. In the plane, the calculator shows the z component as a signed number, which is the oriented area of the parallelogram.

How do I use the cross product to find a plane normal?

If A and B are two non-parallel vectors lying in the plane, A × B is a normal vector to it. The coefficients (a, b, c) of the equation ax + by + cz + d = 0 are the components of that normal.

Does the cross product depend on the order of the vectors?

The magnitude does not, but the direction does: B × A is the opposite of A × B. That is why the parallelogram area uses the magnitude, which has no sign, and the triple product changes sign when you swap two of the vectors.

Next steps

The determinant behind the perpendicular vector

In 3D the cross product is A × B = (a₂b₃ - a₃b₂, a₃b₁ - a₁b₃, a₁b₂ - a₂b₁), the symbolic determinant with i, j, k on the first row. The result is a vector perpendicular to both A and B, pointing by the right-hand rule: the fingers sweep from A to B and the thumb points along A × B. Swapping the order flips the sign, A × B = -(B × A), and the product of a vector with itself or with a multiple of itself is the zero vector. A sanity check: i × j = k, that is, (1, 0, 0) × (0, 1, 0) = (0, 0, 1).

The magnitude |A × B| = |A||B| sin θ is the area of the parallelogram built on A and B, and half of it is the triangle area. For A = (3, 4, 2) and B = (-1, 2, 3) the cross product is (8, -11, 10) and the parallelogram area is √(64 + 121 + 100) = √285 ≈ 16.8819. In 2D only the z component exists, a₁b₂ - a₂b₁, and its absolute value is the area: for (2, 1) and (1, 3) it gives 5, and the triangle has 5/2. This is the same determinant used for triangle area from coordinates in the analytic geometry calculator.

A third vector brings in the triple product [A, B, C] = A · (B × C), the 3 by 3 determinant of the components. The absolute value is the parallelepiped volume and one sixth of it is the tetrahedron volume. For A = (3, 4, 2), B = (-1, 2, 3) and C = (1, 0, 2): B × C = (4, 5, -2) and A · (B × C) = 12 + 20 - 4 = 28, so the parallelepiped has volume 28 and the tetrahedron 14/3. The sign depends on the order and the orientation of the basis. If the triple product is zero the three vectors are coplanar, and the calculator flags that case with a notice.

Checked examples

Standard basis

Input: A = (1, 0, 0), B = (0, 1, 0)
Expected output: A × B = (0, 0, 1)

i × j = k.

Radical in the area

Input: A = (3, 4, 2), B = (-1, 2, 3)
Expected output: A × B = (8, -11, 10); área = √285

Triangle: √285/2.

Perpendicular axes

Input: A = (2, 0, 0), B = (0, 3, 0)
Expected output: A × B = (0, 0, 6); área = 6

2 by 3 rectangle.

2D case

Input: A = (2, 1), B = (1, 3)
Expected output: A × B = 5; triângulo = 5/2

Only the z component.

General vectors

Input: A = (1, 2, 3), B = (4, -5, 6)
Expected output: A × B = (27, 6, -13); área = √934

≈ 30.5614.

Triple product

Input: A = (3, 4, 2), B = (-1, 2, 3), C = (1, 0, 2)
Expected output: [A, B, C] = 28; V = 28; tetraedro = 14/3

Positively oriented basis.

Full tool FAQ

How do you add and subtract vectors?

How do you calculate the magnitude and the unit vector?

How do you calculate the dot product and the angle between two vectors?

How do you calculate the cross product and the parallelogram area?

What is the triple product and what is it for?

How do you project one vector onto another?

How do you convert magnitude and angle into components?

How do you find the resultant of several forces and the equilibrant force?

How can you tell whether two vectors are parallel, orthogonal or coplanar?

Is the result exact? What are the limits?

Frequently asked questions

Why does the cross product only exist in 3D?

How do I use the cross product to find a plane normal?

If A and B are two non-parallel vectors lying in the plane, A × B is a normal vector to it. The coefficients (a, b, c) of the equation ax + by + cz + d = 0 are the components of that normal.

Continue in the tool

More pages for this tool

Cross product, parallelogram area and triple product

The determinant behind the perpendicular vector

Checked examples

Standard basis

Radical in the area

Perpendicular axes

2D case

General vectors

Triple product

Full tool FAQ

Frequently asked questions

Why does the cross product only exist in 3D?

How do I use the cross product to find a plane normal?

Does the cross product depend on the order of the vectors?

Next steps

Sum, magnitude and direction

Products and angle

Projection

Continue in the tool

More pages for this tool

Cross product, parallelogram area and triple product

The determinant behind the perpendicular vector

Checked examples

Standard basis

Radical in the area

Perpendicular axes

2D case

General vectors

Triple product

Full tool FAQ

Frequently asked questions

Why does the cross product only exist in 3D?

How do I use the cross product to find a plane normal?

Does the cross product depend on the order of the vectors?

Next steps

Sum, magnitude and direction

Products and angle

Projection