The problem asks for the magnitude of the sum of two vectors $\vec{a} = \{3, -5, 8\}$ and $\vec{b} = \{-1, 1, -4\}$.

Step 1: Add the vectors

The sum of the vectors $\vec{a} + \vec{b}$ is calculated component-wise: $\vec{a} + \vec{b} = \{3 + (-1), -5 + 1, 8 + (-4)\}$ $\vec{a} + \vec{b} = \{2, -4, 4\}$

Step 2: Find the magnitude of the resulting vector

The magnitude of a vector $\vec{v} = \{x, y, z\}$ is given by: $|\vec{v}| = \sqrt{x^2 + y^2 + z^2}$ For $\vec{v} = \{2, -4, 4\}$, we compute: $|\vec{a} + \vec{b}| = \sqrt{2^2 + (-4)^2 + 4^2}$ $|\vec{a} + \vec{b}| = \sqrt{4 + 16 + 16}$ $|\vec{a} + \vec{b}| = \sqrt{36} = 6$

Final Answer:

The magnitude of the sum of the vectors is: $\boxed{6}$

Let me know if you want further clarification or have any questions! Here are some related questions you might find useful:

1. How is vector addition defined in higher dimensions?
2. What are the properties of vector magnitudes?
3. How can vector magnitudes be applied in physics problems?
4. What are the differences between vector addition and scalar multiplication?
5. How would the result change if the vectors had four components instead of three?

Tip: Always double-check