Find a unit vector in the same direction as the given vector,
→ d

To find a unit vector in the same direction as a given vector $\vec{d} = 3.4 \, \hat{i} + 4.21 \, \hat{j}$, follow these steps:

Step 1: Find the magnitude of the vector

The magnitude of $\vec{d}$ is calculated as: $\|\vec{d}\| = \sqrt{(3.4)^2 + (4.21)^2}$

$\|\vec{d}\| = \sqrt{11.56 + 17.7241}$

$\|\vec{d}\| = \sqrt{29.2841} \approx 5.4122$

Step 2: Find the unit vector

The unit vector $\hat{u}$ is obtained by dividing each component of $\vec{d}$ by its magnitude: $\hat{u} = \frac{\vec{d}}{\|\vec{d}\|}$

$\hat{u} = \left(\frac{3.4}{5.4122}\right) \, \hat{i} + \left(\frac{4.21}{5.4122}\right) \, \hat{j}$

Step 3: Simplify and round

$\frac{3.4}{5.4122} \approx 0.6279, \quad \frac{4.21}{5.4122} \approx 0.7778$

Thus, the unit vector is: $\hat{u} \approx 0.6279 \, \hat{i} + 0.7778 \, \hat{j}$

Final Answer:

$\hat{u} = 0.6279 \, \hat{i} + 0.7778 \, \hat{j}$

Let me know if you have questions or need a detailed breakdown of any step!

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1. How is a unit vector defined, and why is it useful in mathematics and physics?
2. What is the significance of rounding values in calculations, and how does it affect the results?
3. Can the process for finding a unit vector be generalized to three-dimensional vectors?
4. Why is the magnitude of a vector always a positive value?
5. What are some real-world applications of unit vectors?

Tip: Always double-check the magnitude calculation as it ensures correctness throughout the process!