Orthogonal basis from (2, 1, 2), (1, 1, 0)

Step by step

The question

Question: make (2, 1, 2) and (1, 1, 0) orthogonal, with Gram-Schmidt.

We want new vectors u₁ and u₂, each at a right angle to the others, that make the same vectors as the old ones.

The idea: take away from each vector the parts along the earlier ones.

1 Keep the first

The first one stays: u₁ = v₁.

2 Make the second one perpendicular

u₂ is v₂ with its part along u₁ taken away.

Careful!

The projection is taken away from v₂, not added.

The part along u₁, as in the projection lesson.

The top: v₂ • u₁.

The bottom: u₁ • u₁.

So the number is 3/9, made simpler: 1/3.

Times u₁: every entry times 1/3.

The projection is (2/3, 1/3, 2/3).

So u₂ is v₂ minus the projection.

Entry by entry: (1/3, 2/3, −2/3).

To clear the fractions, we multiply it by 3.

Same direction, still at a right angle.

So the orthogonal vectors are u₁ = (2, 1, 2) and u₂ = (1, 2, −2).

3 Check

Let's check: every pair must have dot product 0.

u₁ • u₂.

Every dot product is 0: the vectors are at right angles.

So the answer is right!

Let's look back.

First: keep the first.

Then: make the second one perpendicular.

Last: check.