Calculate <-1, 2, 4> × <3, 0, -2> Step by Step

Step by step

The question

Question: find the cross product of u = ⟨−1, 2, 4⟩ and v = ⟨3, 0, −2⟩.

⟨−1, 2, 4⟩ × ⟨3, 0, −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; −1, 2, 4; 3, 0, −2|

2 The x part

Cover the i column.

Four numbers are left: 2, 4, 0, −2.

Write them as a small grid.

Its cross: down 2 times (−2), minus up 4 times 0.

x part = |2, 4; 0, −2| = 2 × (−2) − 4 × 0

−4 minus 0 is −4.

x part = −4 − 0 = −4

3 The y part

Now cover the j column.

Four numbers are left: −1, 4, 3, −2.

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

Its cross: down −1 times (−2), minus up 4 times 3.

y part = −|−1, 4; 3, −2| = −(−1 × (−2) − 4 × 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

2 minus 12 is −10.

With the minus in front: 10.

y part = −(2 − 12) = 10

4 The z part

Now cover the k column.

Four numbers are left: −1, 2, 3, 0.

Write them as a small grid.

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

z part = |−1, 2; 3, 0| = −1 × 0 − 2 × 3

0 minus 6 is −6.

z part = 0 − 6 = −6

Put the three parts together: u × v = ⟨−4, 10, −6⟩.

u × v = ⟨−4, 10, −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!

⟨−4, 10, −6⟩ • ⟨−1, 2, 4⟩ = 4 + 20 − 24 = 0

And with v: 0.

Right!

⟨−4, 10, −6⟩ • ⟨3, 0, −2⟩ = −12 + 0 + 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