Projections

The shadow one vector casts on another.

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

NameDate

Figure — use these to answer the problems

Imagine the shadow one arrow casts on the other. Turn them perpendicular and the shadow vanishes to nothing.

u, first entry4

u, second entry3

v, first entry5

v, second entry0

Figure 1

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
2
What is left after subtracting the projection of u onto v from u?
- The part of u that is orthogonal to v.
- Always the zero vector.
u = ⟨5, 9⟩ projected onto ⟨1, 0⟩. What is the first entry?
Answer
u = ⟨5, 9⟩ projected onto ⟨0, 1⟩. What is the second entry?
Answer
u · v = 12 and v · v = 4. How many copies of v does the projection take?
Answer

The same ideas with more to keep track of.

3 problems

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

4 problems

9
Match each piece of the decomposition to what it is.
Draw a line from each item on the left to its match on the right.
- The projection of u onto v
- u minus that projection
- The two pieces added together
- The part of u lying along v
- The part of u orthogonal to v
- u itself
u · v = 20 and v · v = 5. How many copies of v does the projection take?
Answer
The Shadow
u = ⟨7, 2⟩ projected onto ⟨1, 0⟩. What is the first entry of the projection?
Answer
The Vanishing
u is orthogonal to v. What is the norm of the projection of u onto v?
Answer