Cross Product Calculator

What is the cross product?

The cross product takes two vectors in three dimensions and gives back a third vector. That new vector has two properties. It stands at right angles to both of the vectors you started with, and its length equals the area of the parallelogram the two vectors make when they start from the same point. It is also called the vector product, because the answer is a vector, unlike the dot product, which gives a single number.

Which of the two perpendicular directions it takes is fixed by the right-hand rule, covered below. If the two vectors point along the same line, they span no area, and the cross product is the zero vector.

The cross product formula

Most textbooks write the same thing as a determinant, with the unit vectors i, j and k in the top row, the components of a in the middle row and the components of b in the bottom row:

The magnitude has a formula of its own, |a × b| = |a| |b| sin θ, where θ is the angle between the two vectors. It is the base times the height of the parallelogram.

How to calculate the cross product

Expand the determinant along the top row. For each of i, j and k, cover its column and cross-multiply the 2 × 2 block that is left: top left times bottom right, minus top right times bottom left. The middle part gets a minus sign in front. For a = ⟨1, 2, 3⟩ and b = ⟨4, 5, 6⟩:

1. i part: cover the i column, leaving 2, 3 over 5, 6. Then 2·6 − 3·5 = 12 − 15 = −3.
2. j part: cover the j column, leaving 1, 3 over 4, 6. Then 1·6 − 3·4 = −6, and with the minus sign in front that is 6.
3. k part: cover the k column, leaving 1, 2 over 4, 5. Then 1·5 − 2·4 = 5 − 8 = −3.

So a × b = ⟨−3, 6, −3⟩, or −3i + 6j − 3k in i, j, k form. Check it with the dot product: ⟨−3, 6, −3⟩ · ⟨1, 2, 3⟩ = −3 + 12 − 9 = 0, so the answer really is at right angles to a.

Show ⟨1, 2, 3⟩ × ⟨4, 5, 6⟩ in the calculator

A shortcut that avoids the minus sign: write the components of each vector twice in a row, a₁ a₂ a₃ a₁ a₂, and read the parts off in a cycle. The first part uses positions 2 and 3, the second uses 3 and 1, and the third uses 1 and 2. Each is "first times second minus second times first".

Cross product in i, j, k form

When the vectors are written with i, j and k, read the numbers in front of each letter as the components. 3i − 2j + k is ⟨3, −2, 1⟩, and i + 4j − 2k is ⟨1, 4, −2⟩. A missing letter means 0. Their cross product is ⟨0, 7, 14⟩, which is 7j + 14k.

Show (3i − 2j + k) × (i + 4j − 2k)

The unit vectors themselves follow a simple pattern. Read the table as row × column:

Going around the cycle i → j → k → i gives a positive answer: i × j = k, j × k = i, k × i = j. Going against it gives the negative, and any unit vector crossed with itself is 0.

The magnitude and the area of a parallelogram

To get the length from the components, square them, add, and take the square root. For ⟨−3, 6, −3⟩ that is √(9 + 36 + 9) = √54 = 3√6, about 7.3485. The calculator keeps the root exact and shows the decimal next to it.

A triangle with sides a and b is half of that parallelogram, so its area is ½ |a × b|. For the vectors above that is 3√6/2, about 3.6742.

Direction and the right-hand rule

Two directions are at right angles to a and b: one on each side of the parallelogram. The cross product takes the one given by your right hand. Point your fingers along a, curl them toward b through the smaller angle, and your thumb points along a × b. With i along the x-axis and j along the y-axis, the thumb points up the z-axis, which is why i × j = k.

Swap the order and your hand turns over, so b × a = −(a × b). The cross product is not commutative, and the order of the rows in the determinant changes the sign of the answer.

Normal vector and plane through three points

Three points that are not on one line fix a plane. To find its normal vector, make two vectors that start at the same point and take their cross product. For P(1, 0, 0), Q(0, 2, 0) and R(0, 0, 3):

1. PQ = Q − P = ⟨−1, 2, 0⟩ and PR = R − P = ⟨−1, 0, 3⟩.
2. The normal is PQ × PR = ⟨6, 3, 2⟩.
3. Put the normal and P into n₁x + n₂y + n₃z = n · P: 6x + 3y + 2z = 6.
4. The triangle PQR has area ½ |PQ × PR| = ½ √49 = 7/2 = 3.5.

Show the plane through these three points

If the cross product comes out as ⟨0, 0, 0⟩, the three points lie on one line. They do not fix a single plane, and the triangle has no area.

Cross product of 2D vectors

The cross product needs three components, but a vector in the plane can be treated as one with z = 0. Then a × b points straight along the z-axis: ⟨a₁, a₂, 0⟩ × ⟨b₁, b₂, 0⟩ = ⟨0, 0, a₁b₂ − a₂b₁⟩. The number a₁b₂ − a₂b₁ is what people mean by the 2D cross product. It is the signed area of the parallelogram: positive when b is counterclockwise from a, negative when it is clockwise. For ⟨3, 4⟩ and ⟨1, 2⟩ it is 3·2 − 4·1 = 2.

Show ⟨3, 4⟩ × ⟨1, 2⟩

Worked examples

Each row was worked out the same way as above. Click a pair to open it in the calculator.

Cross product vs dot product

Use the dot product to measure how much two vectors point the same way, or to test for a right angle. Use the cross product when you need a direction at right angles to two vectors, or an area. The two together give the angle, tan θ = |a × b| ÷ (a · b), which is how the calculator works it out.

Properties of the cross product

• Swapping the order changes the sign: b × a = −(a × b).
• It spreads over addition: a × (b + c) = a × b + a × c.
• A number can be moved out: (ka) × b = k(a × b).
• Any vector crossed with itself, or with a parallel vector, gives 0.
• It is not associative: (a × b) × c is usually not the same as a × (b × c). For example, (i × i) × j = 0, but i × (i × j) = i × k = −j.

Three vectors give one more useful number, the scalar triple product a · (b × c). Its absolute value is the volume of the box with edges a, b and c, and it is 0 when the three vectors lie in one plane. To work it out here, find b × c first, then take its dot product with a.

Common mistakes

• Forgetting the minus sign on the j part. The middle part of the determinant is −(a₁b₃ − a₃b₁), which is a₃b₁ − a₁b₃.
• Swapping a and b. b × a points the opposite way, so check which vector the problem puts first.
• Multiplying matching components, a₁b₁, a₂b₂, a₃b₃. That is the start of the dot product, not the cross product.
• Reading a missing letter wrong. In 2i + k there is no j, so the j component is 0, not 1.
• Giving the area of the parallelogram when the question asks for the triangle. The triangle is half.

Where the cross product is used

In physics, torque is the cross product τ = r × F of the lever arm and the force, and the magnetic force on a moving charge is F = qv × B. 3D graphics and games use cross products to find the normal of each triangle in a model, which decides how light falls on it. Surveyors and engineers use the triangle-area version to measure land and surfaces from coordinates.