Calculate <2, 1, 0> × <0, 3, 1> Step by Step

Step by step

The question

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

⟨2, 1, 0⟩ × ⟨0, 3, 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; 2, 1, 0; 0, 3, 1|

2 The x part

Cover the i column.

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

Write them as a small grid.

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

x part = |1, 0; 3, 1| = 1 × 1 − 0 × 3

1 minus 0 is 1.

x part = 1 − 0 = 1

3 The y part

Now cover the j column.

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

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

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

2 minus 0 is 2.

With the minus in front: −2.

y part = −(2 − 0) = −2

4 The z part

Now cover the k column.

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

Write them as a small grid.

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

z part = |2, 1; 0, 3| = 2 × 3 − 1 × 0

6 minus 0 is 6.

z part = 6 − 0 = 6

Put the three parts together: u × v = ⟨1, −2, 6⟩.

u × v = ⟨1, −2, 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!

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

And with v: 0.

Right!

⟨1, −2, 6⟩ • ⟨0, 3, 1⟩ = 0 − 6 + 6 = 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