What the Cross Product Is
The cross product a × b of two 3D vectors is a new vector. It is perpendicular to both a and b, its direction follows the right-hand rule, and its length equals the area of the parallelogram the two vectors span: |a × b| = |a||b| sin θ.
Josiah Willard Gibbs and Oliver Heaviside separated vector analysis from quaternions in the 1880s, and the cross product has been standard physics notation since.
The Determinant Method
Write a 3×3 determinant whose first row is the unit vectors i, j, k, the second row is a and the third row is b. Expand along the first row, remembering the minus on the j term.
a = (2, 3, 4), b = (5, 6, 7)
- i component: 3·7 − 4·6 = 21 − 24 = −3.
- j component: −(2·7 − 4·5) = −(14 − 20) = 6.
- k component: 2·6 − 3·5 = 12 − 15 = −3.
Answer: a × b = (−3, 6, −3)
Check: (−3, 6, −3) · (2, 3, 4) = −6 + 18 − 12 = 0 and with b it is −15 + 36 − 21 = 0 ✓. The length is √54 = 3√6 ≈ 7.35, so the parallelogram has area 7.35 and the triangle half of that.
Direction and the Right-Hand Rule
Point the fingers of your right hand along a and curl them toward b. Your thumb points along a × b. Swap the order and the thumb flips: b × a = −(a × b). That is the key difference from the dot product, which is symmetric.
| Property | Rule |
|---|---|
| Order | a × b = −(b × a) |
| Parallel vectors | a × b = 0 |
| Unit vectors | i × j = k, j × k = i, k × i = j |
| Distributes | a × (b + c) = a × b + a × c |
Triple Product and Planes
a = (1, 2, 3), b = (0, 1, 4), c = (5, 6, 0)
- b × c = (1·0 − 4·6, 4·5 − 0·0, 0·6 − 1·5) = (−24, 20, −5).
- a · (b × c) = −24 + 40 − 15 = 1.
Answer: Volume = 1 (and the tetrahedron has volume 1/6)
If the triple product is 0 the three vectors are coplanar. For a plane through three points, subtract to get two vectors, take their cross product as the normal n, and write n · (X − P) = 0. The points (1,0,0), (0,2,0), (0,0,3) give 6x + 3y + 2z = 6.
Real-World Uses
- Torque: a wrench of 0.5 m with a 20 N push at a right angle gives r × F = 10 N·m.
- Magnetic force on a moving charge: F = q (v × B).
- Computer graphics: the normal of a triangle decides how light hits it.
- Navigation and robotics: angular velocity and rotation axes.
What We Think Is Overrated, Underrated and Wrong
A Worked Example With Real Numbers
For a = (1, 2, 3) and b = (4, 5, 6): a × b = (2·6 − 3·5, 3·4 − 1·6, 1·5 − 2·4) = (−3, 6, −3). Its length √54 = 7.348 is the area of the parallelogram the vectors span. Swapping the order flips the sign: b × a = (3, −6, 3).
Our Take: What Is Overrated, Underrated and Plain Wrong
Overrated: memorising the i, j, k determinant with sign tricks. Write the three components with the cyclic pattern instead.
Underrated: using the cross product to find a normal vector for a plane through three points.
Plain wrong: expecting a × b = b × a. It equals −(b × a).
Cross Product Mistakes That Cost Marks
- Forgetting the minus sign on the j component.
- Swapping the order and getting the opposite direction.
- Treating the result as a number. It is a vector.
- Using the cross product on 2D vectors without setting z = 0 (the result then points along z).
- Confusing sin θ and cos θ: the cross product uses sin, the dot product uses cos.
Why Students Get Stuck on Cross Products, and What a Good Calculator Should Do
Searching for help with cross products usually starts late at night with one problem that will not work out. The formula looks like a determinant, and the direction of the result is hard to picture. A good calculator fixes that by showing each step, checking its own answer and explaining the idea in plain words. This section covers what students struggle with, what they want, and how this page responds.
In short: students need to know what a × b means, not just compute it, not just see a final number, and a calculator earns trust only when its steps can be checked.
Understand the pain points
What students get stuck on
Most mistakes here are not about arithmetic. They come from one rule applied the wrong way round, which is why seeing every step matters.
- Mixing up the order, since a × b = −(b × a)Cross product of (2,3,4) and (5,6,7)↗
- Sign slips in the middle component of the determinantCross product of (1,0,0) and (0,1,0)↗
- Forgetting that the result only exists in three dimensionsCross product of (2,4,6) and (1,2,3)↗
- Not linking the length to the area of a parallelogram
Understand the frustrations and desires
What they feel, and what they want
Each frustration points to a wish: a clearer step, a built-in check, an explanation in plain words.
- Wants the determinant expansion written out
- Wants the direction explained with the right-hand rule
- Wants area and normal vector from the same input
Build the page around their intent
Each intent has a place on this page
The page follows intent, not features, so each need has a clear starting point.
- I need an answer now: type the problem into the calculator at the top of this page.Cross product of (2,3,4) and (5,6,7)↗
- I need to understand: the sections below explain the idea in plain words and list the common mistakes.
- I need to check my work: every step is shown, so the first line that differs from yours is the one to study.
- I need practice: open the solver, tap + and choose Create Practice Test.
- I need it explained aloud: in the solver, tap + and choose Create Video.