Step by step
The question
Question: find the cross product of u = ⟨1, 2, −3⟩ and v = ⟨2, 4, 1⟩.
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: 2, −3, 4, 1.
Write them as a small grid.
Its cross: down 2 times 1, minus up −3 times 4.
2 minus (−12) is 14.
3 The y part
Now cover the j column.
Four numbers are left: 1, −3, 2, 1.
Write them as a small grid, with a minus in front: the middle part always gets one.
Its cross: down 1 times 1, minus up −3 times 2.
Careful!
The y part, under j, always has a minus in front.
Forgetting it is the most common mistake.
1 minus (−6) is 7.
With the minus in front: −7.
4 The z part
Now cover the k column.
Four numbers are left: 1, 2, 2, 4.
Write them as a small grid.
Its cross: down 1 times 4, minus up 2 times 2.
4 minus 4 is 0.
Put the three parts together: u × v = ⟨14, −7, 0⟩.
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.