Okay, so I need to find the projection of vector a onto vector b. Hmm, let's recall the formula for projecting one vector onto another. I think it's something like (a · b / ||b||²) times b. Right, the projection vector proj_b a is equal to (a · b divided by the magnitude squared of b) multiplied by vector b. 

Wait, the question gives me that a · b = 8. So if I use that, then the projection would be (8 / ||b||²) * b. But I need to make sure I remember the formula correctly. Let me verify. Yeah, the projection formula is indeed (a · b / ||b||²) * b. So the scalar multiplier is the dot product divided by the magnitude squared of b, and then you multiply that scalar by vector b to get the projection vector.

So first, I need to compute ||b||². Vector b is given as [2, 6, 3]. To find its magnitude squared, I square each component and add them up. Let's calculate that. 

First component squared: 2² = 4. Second component squared: 6² = 36. Third component squared: 3² = 9. Adding those together: 4 + 36 + 9. Let's see, 4 + 36 is 40, and 40 + 9 is 49. So ||b||² is 49. That's nice, a perfect square. 

So the scalar multiplier becomes 8 / 49. Therefore, the projection vector proj_b a is (8/49) multiplied by vector b. Let me compute each component of that. 

First component: (8/49)*2 = 16/49. Second component: (8/49)*6 = 48/49. Third component: (8/49)*3 = 24/49. So putting that all together, the projection vector is [16/49, 48/49, 24/49]. 

Wait, let me check if that makes sense. The projection should be a vector in the direction of b. Since a · b is positive (8), the projection should be in the same direction as b. All components of the projection vector are positive, which matches since b's components are all positive. Also, the scalar multiplier is 8/49, which is less than 1, so the projection vector is a scaled version of b, shorter than b itself. That seems right. 

Alternatively, if the dot product was negative, the projection would be in the opposite direction. But since it's positive, direction is the same. 

Let me confirm the formula again. Projection of a onto b is (a · b / ||b||²) * b. Yes, that's correct. So substituting the values, we get 8 divided by 49, multiplied by [2, 6, 3], which gives [16/49, 48/49, 24/49]. 

I think that's the answer. Let me just write it in LaTeX format as requested. The projection vector would be \begin{pmatrix} \frac{16}{49} \\ \frac{48}{49} \\ \frac{24}{49} \end{pmatrix}. 

Hmm, is there any possibility I made a calculation error? Let me verify each step again. 

Calculating ||b||²: 2² + 6² + 3² = 4 + 36 + 9 = 49. Correct. Then 8 divided by 49 is 8/49. Multiplying each component of b by 8/49: 2*(8/49) = 16/49, 6*(8/49) = 48/49, 3*(8/49) = 24/49. All correct. 

Another way to check: Let's compute the length of the projection vector. The length should be |a · b| / ||b||. Wait, the formula for the scalar projection is |a · b| / ||b||, but the vector projection is that scalar times the unit vector in the direction of b. So the vector projection's magnitude is |a · b| / ||b||. Let's check the magnitude of our result. 

Compute sqrt( (16/49)^2 + (48/49)^2 + (24/49)^2 ). Let's compute each square:

(16/49)^2 = 256 / 2401

(48/49)^2 = 2304 / 2401

(24/49)^2 = 576 / 2401

Adding them up: 256 + 2304 + 576 = 256 + 2304 is 2560, plus 576 is 3136. So total is 3136 / 2401. Square root of that is sqrt(3136)/sqrt(2401) = 56 / 49 = 8/7. 

Now, the magnitude of the projection vector should be |a · b| / ||b|| = 8 / 7. Which matches the calculation above. So that's a good check. 

Therefore, the projection vector's magnitude is 8/7, and the direction is the same as vector b. The components we calculated also align with the direction of b. So I think this is correct. 

Thus, the final answer is the vector [16/49, 48/49, 24/49]. Expressed in LaTeX, that's the matrix with those fractions. 

I don't see any mistakes here. The steps make sense, and the checks confirm the result. So I'm confident this is the right answer.
Therefore, the projection of $\mathbf{a}$ onto $\mathbf{b}$ is $\boxed{\begin{pmatrix} \frac{16}{49} \\ \frac{48}{49} \\ \frac{24}{49} \end{pmatrix}}$.
</think>