<Question>
The projection of $\begin{pmatrix} 2 \\ y \\ -5 \end{pmatrix}$ onto $\begin{pmatrix} 1 \\ -2 \\ 1 \end{pmatrix}$ is
\[\frac{5}{6} \begin{pmatrix} 1 \\ -2 \\ 1 \end{pmatrix}.\]Find $y.$
</Question>

<think>
<Plan_1>
Okay, let's see. I need to find the value of y such that when I project the vector (2, y, -5) onto the vector (1, -2, 1), the result is (5/6)(1, -2, 1). Hmm, projections. Alright, let me recall how vector projection works.

First, the formula for the projection of a vector **v** onto another vector **u** is given by ( (**v** ⋅ **u**) / ||**u**||² ) **u**. Right? So that's the scalar projection multiplied by the unit vector in the direction of **u**, but written as a scalar multiple of **u**. In this case, the projection is given as (5/6) times the vector (1, -2, 1). So that means the projection formula should equal that.

Let me write down the vectors. Let **v** be the vector (2, y, -5) and **u** be the vector (1, -2, 1). Then, according to the projection formula, proj_**u**(**v**) = ( (**v** ⋅ **u**) / ||**u**||² ) **u**. And this is supposed to equal (5/6) **u**. Therefore, the scalar coefficient ( (**v** ⋅ **u**) / ||**u**||² ) must be equal to 5/6. So I can set up the equation:

(**v** ⋅ **u**) / ||**u**||² = 5/6

Now, let's compute **v** ⋅ **u**.
That's 2*1 + y*(-2) + (-5)*1 = 2 - 2y -5 = (2 -5) -2y = -3 -2y.

Next, compute ||**u**||².
That's 1² + (-2)² + 1² = 1 +4 +1 =6.

So plugging these into the equation:

(-3 -2y)/6 =5/6

Hmm, okay.
Multiply both sides by 6 to eliminate denominators:

-3 -2y =5

Then, solve for y:

-2y =5 +3=8

So y=8/(-2)= -4.

Wait, that seems straightforward. Let me check if I made any errors.

Let me verify by plugging y=-4 back into the original vector and computing the projection.

<prompt_1.1> Verify the solution by recalculating the projection with y = -4. </prompt_1.1>

But let me double-check the steps in case I missed something.

<prompt_1.2> Double-check the solution by re-evaluating the projection formula and calculations. </prompt_1.2>

Wait, just to be thorough, let me make sure that the projection formula was applied correctly. The projection scalar is ( **v** ⋅ **u** ) divided by ||**u**|| squared, then multiplied by **u**.

<prompt_1.3> Perform a final thorough check of the solution and consider if there are alternative methods to solve the problem. </prompt_1.3>

Alternatively, another way to approach this could be to write out the projection vector component-wise and set each component equal to (5/6) times the corresponding component of **u**. Let me try that method to confirm.

<prompt_1.4> Explore an alternative method by writing the projection component-wise and solving for y. </prompt_1.4>
</Plan_1>

<Execution_1>
<execution_1.1>
Let **v** = (2, -4, -5) and **u** = (1, -2, 1). Compute the dot product **v** ⋅ **u**: 2*1 + (-4)*(-2) + (-5)*1 = 2 +8 -5=5. Then ||**u**||² is 6 as before. So the scalar projection is 5/6, so the projection vector is (5/6)**u**, which matches the given projection.
</execution_1.1>

<execution_1.2>
First, compute **v** ⋅ **u** with y=-4:

(2)(1) + (-4)(-2) + (-5)(1) = 2 +8 -5 =5. Then, ||**u**||² is 6. So 5/6, which is exactly the scalar given. So yes, that checks out.
</execution_1.2>

<execution_1.3>
So, if **v** ⋅ **u** is -3 -2y, and ||**u**||² is 6, so yes, (-3 -2y)/6 is the scalar factor. Then setting that equal to 5/6. So solving -3 -2y =5. Then -2y=8, so y=-4.
</execution_1.3>

<execution_1.4>
So the projection vector is:

( (-3 -2y)/6 ) * (1, -2, 1)

Which should equal (5/6)(1, -2, 1). Therefore, each component must be equal. So for the first component: ( (-3 -2y)/6 ) *1 =5/6. Similarly, the second component: ( (-3 -2y)/6 )*(-2) should equal (5/6)*(-2). The third component: ( (-3 -2y)/6 )*1 =5/6. So all three components must satisfy the equation.

Looking at the first component: (-3 -2y)/6 =5/6. Multiply both sides by 6: -3 -2y=5. Then, as before, -2y=8, so y=-4. Similarly, check the third component: same equation. For the second component: ( (-3 -2y)/6 )*(-2) = (5/6)*(-2). Let's see, left side: (-3 -2y)*(-2)/6. If y=-4, then (-3 -2*(-4))= (-3 +8)=5. Then 5*(-2)/6= -10/6= -5/3. On the right side, (5/6)*(-2)= -10/6= -5/3. So that also checks out.
</execution_1.4>
</Execution_1>

<Plan_2>
Based on execution_1.1, execution_1.2, execution_1.3, execution_1.4:
Therefore, regardless of the component checked, y must be -4. So I think that's solid. No mistakes detected in the reasoning.
**Final Answer**
The value of \( y \) is \(\boxed{-4}\).
</Plan_2>

<Execution_2>
No prompt to execute.
</Execution_2>

<Final_answer>
Therefore, the value of $y$ is $\boxed{-4}$.
</Final_answer>
</think>