Calculate <4, 1, -2> × <1, 3, 1> Step by Step

Step by step

The question

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

⟨4, 1, −2⟩ × ⟨1, 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; 4, 1, −2; 1, 3, 1|

2 The x part

Cover the i column.

Four numbers are left: 1, −2, 3, 1.

Write them as a small grid.

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

x part = |1, −2; 3, 1| = 1 × 1 − −2 × 3

1 minus (−6) is 7.

x part = 1 − (−6) = 7

3 The y part

Now cover the j column.

Four numbers are left: 4, −2, 1, 1.

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

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

y part = −|4, −2; 1, 1| = −(4 × 1 − −2 × 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 (−2) is 6.

With the minus in front: −6.

y part = −(4 − (−2)) = −6

4 The z part

Now cover the k column.

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

Write them as a small grid.

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

z part = |4, 1; 1, 3| = 4 × 3 − 1 × 1

12 minus 1 is 11.

z part = 12 − 1 = 11

Put the three parts together: u × v = ⟨7, −6, 11⟩.

u × v = ⟨7, −6, 11⟩

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!

⟨7, −6, 11⟩ • ⟨4, 1, −2⟩ = 28 − 6 − 22 = 0

And with v: 0.

Right!

⟨7, −6, 11⟩ • ⟨1, 3, 1⟩ = 7 − 18 + 11 = 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