Step by step
The question
Question: find the area of the parallelogram with sides u = ⟨2, 2, 1⟩ and v = ⟨1, −1, 0⟩.
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, 1, −1, 0.
Write them as a small grid.
Its cross: down 2 times 0, minus up 1 times (−1).
0 minus (−1) is 1.
3 The y part
Now cover the j column.
Four numbers are left: 2, 1, 1, 0.
Write them as a small grid, with a minus in front: the middle part always gets one.
Its cross: down 2 times 0, minus up 1 times 1.
Careful!
The y part, under j, always has a minus in front.
Forgetting it is the most common mistake.
0 minus 1 is −1.
With the minus in front: 1.
4 The z part
Now cover the k column.
Four numbers are left: 2, 2, 1, −1.
Write them as a small grid.
Its cross: down 2 times (−1), minus up 2 times 1.
−2 minus 2 is −4.
Put the three parts together: u × v = ⟨1, 1, −4⟩.
5 Find the length
The area of the parallelogram is the length of u × v.
Square the parts, add, take the root.
1 squared is 1.
1 squared is 1.
(−4) squared is 16.
Minus times minus is plus.
Add: 1 plus 1 plus 16 is 18.
But 18 has a square number in it: 9.
18 is 9 times 2.
A root of a times splits into two roots.
√9 is 3.
So ‖u × v‖ = 3√2.
That is the exact answer, made simple.
With a calculator, that is about 4.2, to the nearest tenth.
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 4.2.
4.2 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.