Given the vector $4i-j+8k$, which of the following is a unit vector in the same direction?

• A. $\frac{1}{9}i+\frac{4}{9}j+\frac{8}{9}k$
• B. $\frac{2}{9}i+\frac{1}{9}j+\frac{4}{9}k$
• C. $\frac{4}{17}i-\frac{1}{17}j+\frac{8}{17}k$
• D. $\frac{4}{9}i-\frac{1}{9}j+\frac{8}{9}k$

Answer: (D)

Verified step by step guidance

1. Identify the given vector as \( \vec{v} = 4\mathbf{i} - 1\mathbf{j} + 8\mathbf{k} \).
2. Calculate the magnitude (length) of the vector using the formula: \[ \|\vec{v}\| = \sqrt{4^2 + (-1)^2 + 8^2} \]
3. Simplify the expression inside the square root to find the magnitude: \[ \|\vec{v}\| = \sqrt{16 + 1 + 64} \]
4. Form the unit vector by dividing each component of \( \vec{v} \) by its magnitude: \[ \hat{u} = \frac{1}{\|\vec{v}\|} (4\mathbf{i} - 1\mathbf{j} + 8\mathbf{k}) = \left( \frac{4}{\|\vec{v}\|} \right) \mathbf{i} - \left( \frac{1}{\|\vec{v}\|} \right) \mathbf{j} + \left( \frac{8}{\|\vec{v}\|} \right) \mathbf{k} \]
5. Compare the resulting unit vector components with the given options to identify the correct unit vector in the same direction.