Step by step
The question
Question: find the area of the parallelogram with sides u = ⟨2, 0, 0⟩ and v = ⟨0, 3, 4⟩.
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: 0, 0, 3, 4.
Write them as a small grid.
Its cross: down 0 times 4, minus up 0 times 3.
0 minus 0 is 0.
3 The y part
Now cover the j column.
Four numbers are left: 2, 0, 0, 4.
Write them as a small grid, with a minus in front: the middle part always gets one.
Its cross: down 2 times 4, minus up 0 times 0.
Careful!
The y part, under j, always has a minus in front.
Forgetting it is the most common mistake.
8 minus 0 is 8.
With the minus in front: −8.
4 The z part
Now cover the k column.
Four numbers are left: 2, 0, 0, 3.
Write them as a small grid.
Its cross: down 2 times 3, minus up 0 times 0.
6 minus 0 is 6.
Put the three parts together: u × v = ⟨0, −8, 6⟩.
5 Find the length
The area of the parallelogram is the length of u × v.
Square the parts, add, take the root.
0 squared is 0.
(−8) squared is 64.
6 squared is 36.
Minus times minus is plus.
Add: 0 plus 64 plus 36 is 100.
10 times 10 is 100.
So ‖u × v‖ = 10.
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 10.
10 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.