The projection of u onto v is the part of u that lies along v. Everything perpendicular is discarded.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The shadow picture
Shine a light straight down onto the line through v. The shadow of u on that line is the projection.
The formula
The projection is (u · v divided by v · v) times v. The fraction is how many copies of v to take.
The remainder
Subtracting the projection from u leaves a vector orthogonal to v. Any vector splits into these two pieces.
When v is a unit vector
If v has length 1, the denominator is 1 and the projection is simply (u · v) times v.
Why it matters
Projection is how you find the closest point in a subspace to a given vector. That is the whole idea behind least squares.
The shadow one vector casts
The projection of u onto v is the part of u lying along v. It is the closest point to u on the line through v, which is the property that makes projection useful.
The formula and its parts
The projection is (u · v / v · v) times v. The dot product measures how much of u lies along v, and dividing by v · v normalises for the length of v.
The remainder is perpendicular
Subtracting the projection from u leaves a vector orthogonal to v. Splitting a vector into parallel and perpendicular parts is a standard move in physics and in least squares.
Projecting onto a subspace
The same idea projects onto a plane or higher subspace, giving the closest point in it. That closest-point property is precisely what least squares exploits.
Step 2: Try It Yourself
Tap and try it out.
- Vector a(4, 3) · length 5
- Vector b(5, 0) · length 5
Step 3: Watch an Example
One step at a time.
Watch Nia Cast a Shadow
Nia projects u = ⟨4, 3⟩ onto v = ⟨1, 0⟩.
- Step 1
She computes u · v = 4(1) + 3(0) = 4.
Step 4: Your Turn
Practice makes it stick.
The Shadow
Problem 1 of 2
u = ⟨7, 2⟩ projected onto ⟨1, 0⟩. What is the first entry of the projection?
The Vanishing
Problem 2 of 2
u is orthogonal to v. What is the norm of the projection of u onto v?
Cast the Shadow
1 of 8
u = ⟨5, 9⟩ projected onto ⟨1, 0⟩. What is the first entry?
2 of 8
u = ⟨5, 9⟩ projected onto ⟨0, 1⟩. What is the second entry?
3 of 8
u · v = 12 and v · v = 4. How many copies of v does the projection take?
4 of 8
v is a unit vector and u · v = 6. What is the norm of the projection?
5 of 8
Projecting u onto itself. How many copies of u does it take?
6 of 8
u · v = 0. How many copies of v does the projection take?
7 of 8
Match each piece of the decomposition to what it is.
Tap a card on the left to start.
8 of 8
u · v = 20 and v · v = 5. How many copies of v does the projection take?
Step 5: Quick Check
Show what you know.
Question 1 of 2
u · v = 18 and v · v = 6. How many copies of v does the projection take?
Question 2 of 2
What is left after subtracting the projection of u onto v from u?
What You Learned
- 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.