Projections

The shadow one vector casts on another.

12 problems · about 23 minutes · N-VM.B.4, N-VM.C.11

What this lesson teaches

The student computes projections of one vector onto another and interprets the remainder.

The projection of u onto v is the part of u lying along v, computed as (u · v)/(v · v) times v.
What is left over after subtracting it is orthogonal to v.
Projection finds the closest point in a subspace, which is the basis of least squares.

Straightforward practice. Get the method working first.

5 problems

u · v = 18 and v · v = 6. How many copies of v does the projection take?
Answer 3
Why 3.
What is left after subtracting the projection of u onto v from u?
Answer The part of u that is orthogonal to v.
Why The orthogonal remainder.
u = ⟨5, 9⟩ projected onto ⟨1, 0⟩. What is the first entry?
Answer 5
Why The x-part of u.
u = ⟨5, 9⟩ projected onto ⟨0, 1⟩. What is the second entry?
Answer 9
Why The y-part of u.
u · v = 12 and v · v = 4. How many copies of v does the projection take?
Answer 3
Why 12 divided by 4.

The same ideas with more to keep track of.

3 problems

v is a unit vector and u · v = 6. What is the norm of the projection?
Answer 6
Why The denominator is 1.
Projecting u onto itself. How many copies of u does it take?
Answer 1
Why u · u over u · u.
u · v = 0. How many copies of v does the projection take?
Answer 0
Why The shadow is nothing.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Match each piece of the decomposition to what it is.
Answer The projection of u onto v → The part of u lying along v; u minus that projection → The part of u orthogonal to v; The two pieces added together → u itself
Why The pieces must reassemble into the original.
u · v = 20 and v · v = 5. How many copies of v does the projection take?
Answer 4
Why 20 over 5.
The Shadow: u = ⟨7, 2⟩ projected onto ⟨1, 0⟩. What is the first entry of the projection?
Answer 7
Why 7.
The Vanishing: u is orthogonal to v. What is the norm of the projection of u onto v?
Answer 0
Why 0.