Step by step
The question
Question: find the area of the triangle with corners (2, 0, 0), (0, 2, 0), (0, 0, 1).
1 Two sides as vectors
Draw the triangle.
Two sides from P make two vectors.
Each is end minus start.
u goes from P to Q: u = ⟨−2, 2, 0⟩.
v goes from P to R: v = ⟨−2, 0, 1⟩.
2 Set up the grid
Write a grid.
On top: i, j, k, for the x, y and z directions.
Under them, u, then v.
3 The x part
Cover the i column.
Four numbers are left: 2, 0, 0, 1.
Write them as a small grid.
Its cross: down 2 times 1, minus up 0 times 0.
2 minus 0 is 2.
4 The y part
Now cover the j column.
Four numbers are left: −2, 0, −2, 1.
Write them as a small grid, with a minus in front: the middle part always gets one.
Its cross: down −2 times 1, minus up 0 times (−2).
Careful!
The y part, under j, always has a minus in front.
Forgetting it is the most common mistake.
−2 minus 0 is −2.
With the minus in front: 2.
5 The z part
Now cover the k column.
Four numbers are left: −2, 2, −2, 0.
Write them as a small grid.
Its cross: down −2 times 0, minus up 2 times (−2).
0 minus (−4) is 4.
Put the three parts together: u × v = ⟨2, 2, 4⟩.
6 Find the length
The area of the parallelogram is the length of u × v.
Square the parts, add, take the root.
2 squared is 4.
2 squared is 4.
4 squared is 16.
Add: 4 plus 4 plus 16 is 24.
But 24 has a square number in it: 4.
24 is 4 times 6.
A root of a times splits into two roots.
√4 is 2.
So ‖u × v‖ = 2√6.
That is the exact answer, made simple.
The triangle is half of the parallelogram.
So we take half.
So the exact area is √6.
With a calculator, that is about 2.4.
7 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 half of the two side lengths multiplied, about 3.2.
2.4 fits!
Let's look back.
First: two sides as vectors.
Then: set up the grid.
Then: the x part.
Then: the y part.
Then: the z part.
Then: find the length.
Last: check.