Cross Product Calculator
This cross product calculator multiplies two vectors into a third one, writes the determinant out in full, and then proves the answer is right before showing it to you.
Give it <1, 2, 3> × <4, 5, 6> and back comes (−3, 6, −3). Notice the middle number is positive.
Two Vectors In, One Vector Out
That sentence is the whole of the difference from the dot product, and it is worth fixing in place before anything else.
A dot product returns a number. A cross product returns an arrow — and that arrow stands at right angles to both of the ones you started with, pointing out of the plane they lie in. If your answer is a single value rather than three, you have done the other operation.
The Determinant, Written Out
Put i, j and k across the top and your two vectors underneath:
| i j k |
| 1 2 3 |
| 4 5 6 |
Cover the i column and two rows of two numbers remain: 2, 3 over 5, 6. Multiply down-right, multiply down-left, subtract: (2)(6) − (3)(5) = −3. Repeat for j and for k.
Writing all three rows out looks slow and saves time. Almost everyone who works from memory instead loses a sign somewhere in the middle.
The Minus in the Middle
Here is where the marks go.
The j term is subtracted, not added. Cover the j column in the example and you get (1)(6) − (3)(4) = −6. Apply the minus and the component becomes +6.
Get that wrong and your answer points the wrong way through the plane, which in a physics question means a wheel spinning backwards or a force pushing the opposite way. The page prints the term both before and after the sign so it cannot slip past.
Check It Yourself in One Line
The cross product is perpendicular to both vectors it came from. Which means you can test your own answer without redoing any of it.
Dot the result with each original vector. Both must come to exactly zero:
(−3)(1) + (6)(2) + (−3)(3) = −3 + 12 − 9 = 0
(−3)(4) + (6)(5) + (−3)(6) = −12 + 30 − 18 = 0
Ten seconds, and it catches a dropped sign every single time. This page runs both dot products in whole-number fractions before it prints anything, which is why it can say the answer is right rather than hope so. I have not found another cross product calculator that checks itself at all.
| Question | Dot product | Cross product |
|---|---|---|
| What comes out? | a number | a vector |
| Does order matter? | no | yes, it reverses |
| Zero means | perpendicular | parallel |
| Works in 2D? | yes | only as a flat pair |
Order Reverses Everything
a × b and b × a are opposites. Every component changes sign.
The dot product does not care which vector you write first, so it is easy to assume this one does not either. It does. Swap them and the arrow flips through the plane: (−3, 6, −3) becomes (3, −6, 3), same length, other direction.
In torque that is the difference between tightening a bolt and loosening it, which is not a rounding error. Put a spanner on a nut and the turning effect is the position vector crossed with the force, in that order: r × F, never F × r. Reverse them and the calculation politely tells you the nut is coming undone when it is going on. The same ordering runs through angular momentum, magnetic force on a moving charge, and the normal to a surface in graphics \u2014 anywhere the answer has a handedness, the order is carrying it.
Zero Has Its Own Meaning
⟨1, 2, 3⟩ × ⟨2, 4, 6⟩ gives (0, 0, 0), and nothing has gone wrong.
The second vector is just the first one doubled, so the two lie along the same line. Parallel vectors enclose no parallelogram, so there is no area, and no plane for a perpendicular to stick out of.
Which makes a cross product of zero the neatest proof that two vectors are parallel — the exact mirror of using a dot product of zero to prove they are perpendicular.
The Length Is an Area
|a × b| is a size, and it is also an area — the parallelogram the two vectors span. Half of it is the triangle with those two as sides.
For ⟨1, 2, 3⟩ and ⟨4, 5, 6⟩ the length is 3√6, so the triangle they make has area 3√6/2. Both come free once you have the cross product, and questions ask for them constantly. Left as surds they stay exact; 7.348 is a reading of the first one, not a replacement for it.
How to Use This Cross Product Calculator
Write both vectors bracketed, separated by ×: <1, 2, 3> × <4, 5, 6>. Three components is the usual case. Two are read as lying flat with z taken as zero, and then the answer can only point straight out of the page, which is exactly what a flat pair should give.
The band gives the vector, the confirmation that it is perpendicular to both, the two areas, and b × a beside it. Show the working opens the determinant, the expansion with the middle sign called out, and the two dot products that prove the result.
If what you wanted was the number rather than the vector, that is the dot product, and the angle between the same two vectors is on the angle between two vectors page. The length of the result here is found the same way as any other vector magnitude.
Cross Product Calculator FAQ
How do you find the cross product of two vectors?
What is the cross product of (1,2,3) and (4,5,6)?
Is the cross product a vector or a number?
How do you check a cross product is right?
Why does the j component have a minus sign?
Is a × b the same as b × a?
What does it mean if the cross product is zero?
What is the magnitude of the cross product used for?
Write the determinant, subtract the middle one, then dot your answer back to prove it.