Chatgpt disagrees: Great question — you're close, but the change you've made is not correct from a geometric and mathematical standpoint.

Let’s clarify:

✅ Original (correct) version:
You can see that this set of points is exactly the set of points for which the projection onto
𝑎
a is the point
𝐶
⋅
𝑎
∣
𝑎
∣
C⋅
∣a∣
a
​
,
and they form a hyperplane orthogonal to
𝑎
a.

This is correct, because:

The projection length is constant (equal to
𝐶
C), so the actual point on the line of
𝑎
a is at a distance of
𝐶
C units along the direction of
𝑎
a.

The unit vector in the direction of
𝑎
a is
𝑎
∣
𝑎
∣
∣a∣
a
​
, so scaling it by
𝐶
C gives you the fixed projection point.

❌ Your modified version:
... the projection onto
𝑎
a is the point
𝐶
⋅
𝑎
∣
𝑎
∣
2
C⋅
∣a∣
2

a
​

This is incorrect, because:

𝑎
∣
𝑎
∣
2
∣a∣
2

a
​
is not a unit vector — it's a scaled-down version of
𝑎
a.

This form appears in a different context: when you’re projecting a vector onto another vector, the formula for projection is:

proj
𝑎
(
𝑟
)

𝑟
⋅
𝑎
∣
𝑎
∣
2
⋅
𝑎
proj
a
​
(r)=
∣a∣
2

r⋅a
​
⋅a
But here you're describing the set of vectors whose dot product with
𝑎
a is constant, not projecting a specific vector onto another.

✅ TL;DR:
Keep the original form:

𝐶
⋅
𝑎
∣
𝑎
∣
C⋅
∣a∣
a
​

It is geometrically meaningful and correct — describing a point lying at a fixed distance
𝐶
C along the direction of
𝑎
a, which is how the hyperplane is constructed.

Let me know if you'd like a visual explanation of this with an example or diagram.

You said:
Yes

ChatGPT said:
Great! Let’s walk through a visual explanation of the statement:

“The set of points
𝑟
r such that
𝑟
⋅
𝑎

𝐶
r⋅a=C form a hyperplane orthogonal to
𝑎
a, and the projection of any such
𝑟
r onto
𝑎
a is the point
𝐶
⋅
𝑎
∣
𝑎
∣
C⋅
∣a∣
a
​
.”

🧠 Step-by-step Visual Intuition:

1. What does
   𝑟
   ⋅
   𝑎
   =
   𝐶
   r⋅a=C mean?
   The dot product
   𝑟
   ⋅
   𝑎
   r⋅a is:

∣
𝑟
∣
⋅
∣
𝑎
∣
⋅
cos
⁡
𝜃
∣r∣⋅∣a∣⋅cosθ
where
𝜃
θ is the angle between
𝑟
r and
𝑎
a.

So fixing
𝑟
⋅
𝑎

𝐶
r⋅a=C means all such vectors
𝑟
r have the same projection onto
𝑎
a.

2. Geometric meaning of constant projection
   Imagine you have a vector
   𝑎
   a, and you're projecting various points in space (vectors
   𝑟
   r) onto the direction of
   𝑎
   a. If all those projections land at the same spot on the line of
   𝑎
   a, those points form a flat surface (or line, in 2D) that is perpendicular to
   𝑎
   a.

That surface is called a hyperplane orthogonal to
𝑎
a.

3. Where does the point
   𝐶
   ⋅
   𝑎
   ∣
   𝑎
   ∣
   C⋅
   ∣a∣
   a
   ​
   come from?
   This is the actual point on the vector
   𝑎
   a's line that is:

Distance
𝐶
C away in the direction of
𝑎
a (since
𝑎
a is a vector),

Scaled by the unit vector in that direction:
𝑎
∣
𝑎
∣
∣a∣
a
​

So:

Projection point

𝐶
⋅
𝑎
∣
𝑎
∣
Projection point=C⋅
∣a∣
a
​

This point is where all the projections of
𝑟
r's in the set fall. It's the foot of the perpendicular from the hyperplane onto the vector
𝑎
a.

📐 Diagram (conceptual)
pgsql
Copy code
*
/ | <-- vector r
/ |
/ |
/ | <-- All these r's project to the same spot
/ |
------------------------>
^ ^
origin projection point = C * (a / |a|)

           -->         direction of vector a

✅ Summary
The correct projection point is:

𝐶
⋅
𝑎
∣
𝑎
∣
C⋅
∣a∣
a
​

because it describes a fixed location on the line in direction of
𝑎
a.

The set of all vectors
𝑟
r with dot product
𝑟
⋅
𝑎

𝐶
r⋅a=C form a hyperplane orthogonal to
𝑎
a, and all of them project onto the same point.

Let me know if you'd like an actual drawn diagram or want to see this in code or coordinate form!

Substituting $r = C \frac{a}{|a|^2}$ and from $a \cdot a = |a|^2$, we see that, indeed, $r \cdot a = C\frac{|a|^2}{|a|^2}=C$, so it actually should be $|a|^2$.

@mhayter please don't take ChatGPT output for granted, it can often spill wrong information.

Note: ChatGPT 5 gives correct expression for it.