Step by step
The question
Question: make (3, 4) and (1, 0) orthogonal, with Gram-Schmidt.
We want new vectors u₁ and u₂, each at a right angle to the others, that still make every vector the old ones make.
The idea: take away from each vector its parts along the earlier ones.
What is left has no part along them, so it is at a right angle to them.
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 3/25.
Times u₁: every entry times 3/25.
The projection is (9/25, 12/25).
So u₂ is v₂ minus the projection.
Entry by entry: (16/25, −12/25).
To clear the fractions, we multiply it by 25, the bottom.
A number times a vector keeps its direction, so it is still at a right angle.
Every entry has the factor 4.
To make the numbers smaller, we divide it by 4.
Same direction again.
So the orthogonal vectors are u₁ = (3, 4) and u₂ = (4, −3).
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.