Calculate <1, 2, 3> × <4, 5, 6> Step by Step

Step by step

The question

Question: find the cross product of u = ⟨1, 2, 3⟩ and v = ⟨4, 5, 6⟩.

⟨1, 2, 3⟩ × ⟨4, 5, 6⟩

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, 2, 3; 4, 5, 6|

2 The x part

Cover the i column.

Four numbers are left: 2, 3, 5, 6.

Write them as a small grid.

Its cross: down 2 times 6, minus up 3 times 5.

x part = |2, 3; 5, 6| = 2 × 6 − 3 × 5

12 minus 15 is −3.

x part = 12 − 15 = −3

3 The y part

Now cover the j column.

Four numbers are left: 1, 3, 4, 6.

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

Its cross: down 1 times 6, minus up 3 times 4.

y part = −|1, 3; 4, 6| = −(1 × 6 − 3 × 4)

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

6 minus 12 is −6.

With the minus in front: 6.

y part = −(6 − 12) = 6

4 The z part

Now cover the k column.

Four numbers are left: 1, 2, 4, 5.

Write them as a small grid.

Its cross: down 1 times 5, minus up 2 times 4.

z part = |1, 2; 4, 5| = 1 × 5 − 2 × 4

5 minus 8 is −3.

z part = 5 − 8 = −3

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

u × v = ⟨−3, 6, −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!

⟨−3, 6, −3⟩ • ⟨1, 2, 3⟩ = −3 + 12 − 9 = 0

And with v: 0.

Right!

⟨−3, 6, −3⟩ • ⟨4, 5, 6⟩ = −12 + 30 − 18 = 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