<Question>
Find the projection of $\mathbf{a}$ onto $\mathbf{b} = \begin{pmatrix} 2 \\ 6 \\ 3 \end{pmatrix}$ if $\mathbf{a} \cdot \mathbf{b} = 8.$
</Question>

<think>
<Plan_1>
Okay, so I need to find the projection of vector a onto vector b. Hmm, let me recall what the projection formula is. I think the projection of a onto b is given by (a · b / ||b||²) * b. Right? So that's the formula. Let me make sure I remember it correctly. The projection is a vector in the direction of b, scaled by the component of a in the direction of b.

The problem states that a · b = 8. That's the dot product. And they want the projection of a onto b. So according to the formula, I need two things: the dot product of a and b, which is given as 8, and the magnitude squared of b. Then I can plug those into the formula.
So first, let me compute ||b||². The vector b is given as [2, 6, 3]. The magnitude squared is just the sum of the squares of its components.

Calculating ||b||²: 2² + 6² + 3². That's 4 + 36 + 9. Let me add those up.

<prompt_1.1> Calculate the magnitude squared of vector b using the components provided in the above plan. </prompt_1.1>
-----

But let me check if there's another way to interpret the question. Wait, sometimes people might ask for the scalar projection instead of the vector projection. But the question specifically says "the projection of a onto b", and in vector terms, that refers to the vector. The scalar projection would just be the magnitude of that vector, which is comp_b a = (a · b)/||b||. But since they didn't specify scalar projection, I think it's safe to assume it's the vector projection.
-----

But let me check once more if I interpreted the question correctly. The problem states: "Find the projection of a onto b = [2,6,3] if a · b = 8". So they give b and the dot product, and want the projection.
Since the projection formula only requires the dot product and the magnitude squared of b, which I computed as 49, and the dot product is 8, then yes, 8/49 times b is correct.
-----

Is there a possibility that the question expects a different form? For example, maybe they want the answer in terms of the original vector b, but scaled? But 8/49 times b is already in terms of b.
Alternatively, if they had asked for the scalar projection, it would be 8 divided by the magnitude of b, which is 7, so 8/7.
But since they asked for the projection vector, the answer is (8/49)b.
-----

Wait, but let me confirm the formula once again. The vector projection of a onto b is indeed (a · b / ||b||²) b.
Yes, that's correct. So substituting the values, 8 divided by 49, multiplied by the vector b.
So the answer is (8/49) times [2,6,3], which is [16/49, 48/49, 24/49].

<prompt_1.2> Confirm the formula and calculations one last time to ensure accuracy. </prompt_1.2>
</Plan_1>

<Execution_1>
<execution_1.1>
4 + 36 is 40, then 40 + 9 is 49. Oh, that's convenient. So ||b||² is 49.
</execution_1.1>

<execution_1.2>
Therefore, I think that's the solution. Let me just verify once more:

Given a · b = 8, and ||b||² = 49. Then proj_b a = (8/49) b. So yes, that's the calculation. The components are multiplied by 8/49, so 2*(8/49) = 16/49, 6*(8/49)=48/49, 3*(8/49)=24/49. All correct.
</execution_1.2>
</Execution_1>

<Plan_2>
Based on execution_1.1:
So then the projection should be (8 / 49) times the vector b, which is [2, 6, 3].
So multiplying each component by 8/49, right?

Let me compute that. So the projection vector would be:

<prompt_2.1> Compute the projection vector by multiplying each component of b by 8/49. </prompt_2.1>
-----

Wait, but let me make sure that the formula is indeed (a · b / ||b||²) b. Sometimes different sources might present it differently, but I think this is correct.
Alternatively, the formula can be written as (a · b / ||b||²) times the vector b. Yes, that's the same thing.
-----

Alternatively, if they had asked for the scalar projection, it would have been (a · b) / ||b||, which is 8 / 7, but since they asked for the vector projection, the answer is (8/49)b.
-----

Another way to check: if I compute the magnitude of the projection vector, it should be |(a · b)| / ||b||, which is 8 / 7. Let's check the magnitude of [16/49, 48/49, 24/49].
The magnitude would be sqrt( (16/49)^2 + (48/49)^2 + (24/49)^2 ) = (1/49) sqrt(16² + 48² + 24²).

<prompt_2.2> Calculate the magnitude of the projection vector and verify it matches the expected value. </prompt_2.2>
-----

I think that's thorough.
I checked the formula, computed the magnitude squared of b, multiplied b by the scalar factor 8/49, and even verified the magnitude of the resulting vector. All steps seem correct.
</Plan_2>

<Execution_2>
<execution_2.1>
First component: 2 * (8/49) = 16/49

Second component: 6 * (8/49) = 48/49

Third component: 3 * (8/49) = 24/49

So putting it all together, the projection vector is [16/49, 48/49, 24/49].
</execution_2.1>

<execution_2.2>
Calculating 16²: 256, 48²: 2304, 24²: 576. Adding those: 256 + 2304 = 2560, 2560 + 576 = 3136. sqrt(3136) is 56. So 56/49 = 8/7. Which matches the expected scalar projection. So that's a good consistency check. The magnitude of the projection vector is 8/7, which is the scalar projection.
</execution_2.2>
</Execution_2>

<Final_answer>
Therefore, the projection of $\mathbf{a}$ onto $\mathbf{b}$ is $\boxed{\begin{pmatrix} \frac{16}{49} \\ \frac{48}{49} \\ \frac{24}{49} \end{pmatrix}}$.
</Final_answer>
</think>