Step-by-Step Solution:

We are tasked with finding which vector w should be added to $\mathbf{u} + 3\mathbf{v}$ to yield the unit vector $\mathbf{r} = \langle 1, 0 \rangle$. Let's calculate this systematically.

Step 1: Compute $\mathbf{u} + 3\mathbf{v}$

The given vectors are: $\mathbf{u} = \langle -7, 6 \rangle, \quad \mathbf{v} = \langle -4, 17 \rangle.$ First, calculate $3\mathbf{v}$: $3\mathbf{v} = 3 \cdot \langle -4, 17 \rangle = \langle -12, 51 \rangle.$

Add $\mathbf{u}$ and $3\mathbf{v}$: $\mathbf{u} + 3\mathbf{v} = \langle -7, 6 \rangle + \langle -12, 51 \rangle = \langle -19, 57 \rangle.$

Step 2: Solve for $\mathbf{w}$

The resultant vector after adding $\mathbf{w}$ is $\mathbf{r} = \langle 1, 0 \rangle$. This gives the equation: $\mathbf{u} + 3\mathbf{v} + \mathbf{w} = \mathbf{r}.$

Rearrange to solve for $\mathbf{w}$: $\mathbf{w} = \mathbf{r} - (\mathbf{u} + 3\mathbf{v}).$

Substitute $\mathbf{r} = \langle 1, 0 \rangle$ and $\mathbf{u} + 3\mathbf{v} = \langle -19, 57 \rangle$: $\mathbf{w} = \langle 1, 0 \rangle - \langle -19, 57 \rangle = \langle 1 + 19, 0 - 57 \rangle = \langle 20, -57 \rangle.$

Step 3: Identify the correct option

The computed $\mathbf{w} = \langle 20, -57 \rangle$ matches Option C.

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Final Answer:

$\boxed{\text{C. } \mathbf{w} = \langle 20, -57 \rangle}$

Would you like further details or clarifications? Here are some related questions to deepen your understanding:

1. How do you find a unit vector in a given direction?
2. What is the significance of the dot product in vector operations?
3. How can vector components be used to solve physical problems?
4. How would this problem change if $\mathbf{r}$ were a different unit vector?
5. Can you explain how scalar multiplication affects vector magnitudes?

Tip: Always verify vector calculations by rechecking component-wise arithmetic.