Area of the triangle with corners (0, 0, 0), (2, 1, 2), (1, 2, -2) Step by Step

Step by step

The question

Question: find the area of the triangle with corners (0, 0, 0), (2, 1, 2), (1, 2, −2).

P(0, 0, 0), Q(2, 1, 2), R(1, 2, −2)

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, 1, 2⟩.

u = PQ = ⟨2, 1, 2⟩

v goes from P to R: v = ⟨1, 2, −2⟩.

v = PR = ⟨1, 2, −2⟩

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.

u × v = |i, j, k; 2, 1, 2; 1, 2, −2|

3 The x part

Cover the i column.

Four numbers are left: 1, 2, 2, −2.

Write them as a small grid.

Its cross: down 1 times (−2), minus up 2 times 2.

x part = |1, 2; 2, −2| = 1 × (−2) − 2 × 2

−2 minus 4 is −6.

x part = −2 − 4 = −6

4 The y part

Now cover the j column.

Four numbers are left: 2, 2, 1, −2.

Write them as a small grid, with a minus in front: the middle part always gets one.

Its cross: down 2 times (−2), minus up 2 times 1.

y part = −|2, 2; 1, −2| = −(2 × (−2) − 2 × 1)

Careful!

The y part, under j, always has a minus in front.

Forgetting it is the most common mistake.

Careful: the middle part gets a minus

−4 minus 2 is −6.

With the minus in front: 6.

y part = −(−4 − 2) = 6

5 The z part

Now cover the k column.

Four numbers are left: 2, 1, 1, 2.

Write them as a small grid.

Its cross: down 2 times 2, minus up 1 times 1.

z part = |2, 1; 1, 2| = 2 × 2 − 1 × 1

4 minus 1 is 3.

z part = 4 − 1 = 3

Put the three parts together: u × v = ⟨−6, 6, 3⟩.

u × v = ⟨−6, 6, 3⟩

6 Find the length

area = ‖u × v‖

The area of the parallelogram is the length of u × v.

Square the parts, add, take the root.

‖u × v‖ = √(−6)² + 6² + 3²

(−6) squared is 36.

6 squared is 36.

3 squared is 9.

Minus times minus is plus.

‖u × v‖ = √36 + 36 + 9

Add: 36 plus 36 plus 9 is 81.

‖u × v‖ = √81

9 times 9 is 81.

9 × 9 = 81

So ‖u × v‖ = 9.

‖u × v‖ = 9

The triangle is half of the parallelogram.

So we take half.

A = 12 × 9

Half of 9 is 9/2.

A = 92

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!

(u × v) • u = 0, (u × v) • v = 0

And the area is at most half of the two side lengths multiplied, about 4.5.

4.5 fits!

area ≤ ½ × ‖u‖ × ‖v‖ ≈ 4.5

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.

Answer