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

Step by step

The question

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

⟨1, −1, 2⟩ × ⟨3, 0, 1⟩

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; 1, −1, 2; 3, 0, 1|

2 The x part

Cover the i column.

Four numbers are left: −1, 2, 0, 1.

Write them as a small grid.

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

x part = |−1, 2; 0, 1| = −1 × 1 − 2 × 0

−1 minus 0 is −1.

x part = −1 − 0 = −1

3 The y part

Now cover the j column.

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

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

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

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

1 minus 6 is −5.

With the minus in front: 5.

y part = −(1 − 6) = 5

4 The z part

Now cover the k column.

Four numbers are left: 1, −1, 3, 0.

Write them as a small grid.

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

z part = |1, −1; 3, 0| = 1 × 0 − −1 × 3

0 minus (−3) is 3.

z part = 0 − (−3) = 3

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

u × v = ⟨−1, 5, 3⟩

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!

⟨−1, 5, 3⟩ • ⟨1, −1, 2⟩ = −1 − 5 + 6 = 0

And with v: 0.

Right!

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