Final answer:
To find the unit vector in the same direction as the given vector 8i−j+4k, calculate the magnitude and divide each vector component by this magnitude to give (8/9)i - (1/9)j + (4/9)k.
Step-by-step explanation:
To determine the unit vector that aligns with the direction of the vector 8i−j+4k, you need to calculate the magnitude of the original vector and then divide each component by this magnitude. The magnitude (or length) of a vector is found by taking the square root of the sum of the squares of its components. For the given vector, the magnitude is \( \sqrt{8^2 + (-1)^2 + 4^2} = \sqrt{64 + 1 + 16} = \sqrt{81} = 9 \).
Next, you divide each component of the vector by its magnitude to get the unit vector in the same direction:
- X component: 8/9
- Y component: -1/9
- Z component: 4/9
Thus, the unit vector is (8/9)i - (1/9)j + (4/9)k.