Step by step
The question
Question: find the area of the parallelogram with sides u = ⟨3, −1, 2⟩ and v = ⟨1, 2, −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: −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.
3 The y part
Now cover the j column.
Four numbers are left: 3, 2, 1, −1.
Write them as a small grid, with a minus in front: the middle part always gets one.
Its cross: down 3 times (−1), minus up 2 times 1.
Careful!
The y part, under j, always has a minus in front.
Forgetting it is the most common mistake.
−3 minus 2 is −5.
With the minus in front: 5.
4 The z part
Now cover the k column.
Four numbers are left: 3, −1, 1, 2.
Write them as a small grid.
Its cross: down 3 times 2, minus up −1 times 1.
6 minus (−1) is 7.
Put the three parts together: u × v = ⟨−3, 5, 7⟩.
5 Find the length
The area of the parallelogram is the length of u × v.
Square the parts, add, take the root.
(−3) squared is 9.
5 squared is 25.
7 squared is 49.
Minus times minus is plus.
Add: 9 plus 25 plus 49 is 83.
83 is not a square number.
So √83 is the exact length.
With a calculator, that is about 9.1, 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 9.2.
9.1 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.