Cross product of <0, 2, -1> and <3, 1, 2> Step by Step

Step by step

The question

Question: find the cross product of u = ⟨0, 2, −1⟩ and v = ⟨3, 1, 2⟩.

⟨0, 2, −1⟩ × ⟨3, 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; 0, 2, −1; 3, 1, 2|

2 The x part

Cover the i 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.

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

4 minus (−1) is 5.

x part = 4 − (−1) = 5

3 The y part

Now cover the j column.

Four numbers are left: 0, −1, 3, 2.

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

Its cross: down 0 times 2, minus up −1 times 3.

y part = −|0, −1; 3, 2| = −(0 × 2 − −1 × 3)

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

0 minus (−3) is 3.

With the minus in front: −3.

y part = −(0 − (−3)) = −3

4 The z part

Now cover the k column.

Four numbers are left: 0, 2, 3, 1.

Write them as a small grid.

Its cross: down 0 times 1, minus up 2 times 3.

z part = |0, 2; 3, 1| = 0 × 1 − 2 × 3

0 minus 6 is −6.

z part = 0 − 6 = −6

Put the three parts together: u × v = ⟨5, −3, −6⟩.

u × v = ⟨5, −3, −6⟩

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, −6⟩ • ⟨0, 2, −1⟩ = 0 − 6 + 6 = 0

And with v: 0.

Right!

⟨5, −3, −6⟩ • ⟨3, 1, 2⟩ = 15 − 3 − 12 = 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