Question

Write the unit vector in the opposite direction to \( \vec{u} = 8\hat{i} + 3\hat{j} - \hat{k} \).

Hint:

The unit vector in the opposite direction is obtained by negating the unit vector in the original direction.

Answer 1

Step 1: Recall the formula for unit vectors.
A unit vector in the direction of a vector \( \vec{v} \) is given by: \[ \hat{v} = \frac{\vec{v}}{|\vec{v}|} \]

Step 2: Find the magnitude of \( \vec{u} \).
The magnitude of \( \vec{u} = 8\hat{i} + 3\hat{j} - \hat{k} \) is: \[ |\vec{u}| = \sqrt{8^2 + 3^2 + (-1)^2} = \sqrt{64 + 9 + 1} = \sqrt{74} \]

Step 3: Find the unit vector in the opposite direction.
The unit vector in the opposite direction is: \[ \hat{u} = -\frac{\vec{u}}{|\vec{u}|} = -\frac{8\hat{i} + 3\hat{j} - \hat{k}}{\sqrt{74}} = -\frac{8}{\sqrt{74}}\hat{i} - \frac{3}{\sqrt{74}}\hat{j} + \frac{1}{\sqrt{74}}\hat{k} \]

Final Answer: \[ \boxed{-\frac{8}{\sqrt{74}}\hat{i} - \frac{3}{\sqrt{74}}\hat{j} + \frac{1}{\sqrt{74}}\hat{k}} \]