Step by step
The question
Question: find the cross product of u = ⟨2, −3, 1⟩ and v = ⟨1, 2, −2⟩.
1 Set up the grid
Write a grid.
On top: i, j, k, for the x, y and z directions.
Under them, u, then v.
2 The x part
Cover the i column.
Four numbers are left: −3, 1, 2, −2.
Write them as a small grid.
Its cross: down −3 times (−2), minus up 1 times 2.
6 minus 2 is 4.
3 The y part
Now cover the j column.
Four numbers are left: 2, 1, 1, −2.
Write them as a small grid, with a minus in front: the middle part always gets one.
Its cross: down 2 times (−2), minus up 1 times 1.
Careful!
The y part, under j, always has a minus in front.
Forgetting it is the most common mistake.
−4 minus 1 is −5.
With the minus in front: 5.
4 The z part
Now cover the k column.
Four numbers are left: 2, −3, 1, 2.
Write them as a small grid.
Its cross: down 2 times 2, minus up −3 times 1.
4 minus (−3) is 7.
Put the three parts together: u × v = ⟨4, 5, 7⟩.
5 Check
Let's check.
The answer must be at a square corner to u, so its dot product with u is 0.
It is 0!
And with v: 0.
Right!
Here is a picture of the directions.
Each arrow is drawn the same length.
u and v lie in this flat piece.
u × v sticks straight out of it.
Let's look back.
First: set up the grid.
Then: the x part.
Then: the y part.
Then: the z part.
Last: check.