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

Step by step

The question

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

We want new vectors u₁, u₂ and u₃, 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.

The part along u₁, as in the projection lesson.

The top: v₂ • u₁.

The bottom: u₁ • u₁.

So the number is 2/3.

Times u₁: every entry times 2/3.

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

So u₂ is v₂ minus the projection.

Entry by entry: (1/3, 1/3, −2/3).

To clear the fractions, we multiply it by 3.

Same direction, still at a right angle.

3 Make the third one perpendicular

u₃ is v₃ with its parts along u₁ and u₂ taken away.

Careful!

We project onto the new vectors u₁ and u₂, not onto the old ones.

The part along u₁, as in the projection lesson.

The top: v₃ • u₁.

The bottom: u₁ • u₁.

So the number is 1/3.

Times u₁: every entry times 1/3.

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

Now the part along u₂.

The top: v₃ • u₂.

The bottom: u₂ • u₂.

So the number is 1/6.

Times u₂: every entry times 1/6.

The projection is (1/6, 1/6, −1/3).

So u₃ is v₃ minus both projections.

Entry by entry: (1/2, −1/2, 0).

To clear the fractions, we multiply it by 2.

Same direction, still at a right angle.

So the orthogonal vectors are u₁ = (1, 1, 1), u₂ = (1, 1, −2) and u₃ = (1, −1, 0).

4 Check

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

u₁ • u₂.

u₁ • u₃.

u₂ • u₃.

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.

Then: make the third one perpendicular.

Last: check.