Step by step
The question
Question: find the area of the parallelogram with sides u = ⟨1, 2, 2⟩ and v = ⟨2, 1, −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: 2, 2, 1, −2.
Write them as a small grid.
Its cross: down 2 times (−2), minus up 2 times 1.
−4 minus 2 is −6.
3 The y part
Now cover the j column.
Four numbers are left: 1, 2, 2, −2.
Write them as a small grid, with a minus in front: the middle part always gets one.
Its cross: down 1 times (−2), minus up 2 times 2.
Careful!
The y part, under j, always has a minus in front.
Forgetting it is the most common mistake.
−2 minus 4 is −6.
With the minus in front: 6.
4 The z part
Now cover the k column.
Four numbers are left: 1, 2, 2, 1.
Write them as a small grid.
Its cross: down 1 times 1, minus up 2 times 2.
1 minus 4 is −3.
Put the three parts together: u × v = ⟨−6, 6, −3⟩.
5 Find the length
The area of the parallelogram is the length of u × v.
Square the parts, add, take the root.
(−6) squared is 36.
6 squared is 36.
(−3) squared is 9.
Minus times minus is plus.
Add: 36 plus 36 plus 9 is 81.
9 times 9 is 81.
So ‖u × v‖ = 9.
6 Check
Let's check the cross product: it is at a square corner to u and to v.
Both dot products are 0.
Right!
And the area is at most the two side lengths multiplied, about 9.
9 fits!
Let's look back.
First: set up the grid.
Then: the x part.
Then: the y part.
Then: the z part.
Then: find the length.
Last: check.