Calculate <2, 3, 1> × <1, 1, 2> Step by Step

Step by step

The question

Question: find the cross product of u = ⟨2, 3, 1⟩ and v = ⟨1, 1, 2⟩.

⟨2, 3, 1⟩ × ⟨1, 1, 2⟩

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.

u × v = |i, j, k; 2, 3, 1; 1, 1, 2|

2 The x part

Cover the i 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.

x part = |3, 1; 1, 2| = 3 × 2 − 1 × 1

6 minus 1 is 5.

x part = 6 − 1 = 5

3 The y part

Now cover the j column.

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

y part = −|2, 1; 1, 2| = −(2 × 2 − 1 × 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 1 is 3.

With the minus in front: −3.

y part = −(4 − 1) = −3

4 The z part

Now cover the k column.

Four numbers are left: 2, 3, 1, 1.

Write them as a small grid.

Its cross: down 2 times 1, minus up 3 times 1.

z part = |2, 3; 1, 1| = 2 × 1 − 3 × 1

2 minus 3 is −1.

z part = 2 − 3 = −1

Put the three parts together: u × v = ⟨5, −3, −1⟩.

u × v = ⟨5, −3, −1⟩

5 Check

Let's check.

The answer must be at a square corner to u, so its dot product with u is 0.

It is 0!

⟨5, −3, −1⟩ • ⟨2, 3, 1⟩ = 10 − 9 − 1 = 0

And with v: 0.

Right!

⟨5, −3, −1⟩ • ⟨1, 1, 2⟩ = 5 − 3 − 2 = 0

Here is a picture of the directions.

Each arrow is drawn the same length.

u and v lie in this flat piece.

u × v sticks straight out of it.

Let's look back.

First: set up the grid.

Then: the x part.

Then: the y part.

Then: the z part.

Last: check.

Answer