https://file.aiquickdraw.com/mathbot/file/1727194125481pp39low8.jpg

Let's solve this step by step. The displacement vector is given as:

\[
\Delta \vec{r} = -8.00\, \hat{i} + 17.0\, \hat{j} \, [m]
\]

Where:
- The \( \hat{i} \) direction represents the x-axis (East-West direction).
- The \( \hat{j} \) direction represents the y-axis (North-South direction).

We need to find the displacement in magnitude-bearing form. Here's how we proceed:

### Step 1: Calculate the magnitude
The magnitude \( |\Delta \vec{r}| \) of the displacement vector is found using the Pythagorean theorem:

\[
|\Delta \vec{r}| = \sqrt{(-8.00)^2 + (17.0)^2}
\]

Let's compute that:

\[
|\Delta \vec{r}| = \sqrt{64 + 289} = \sqrt{353} \approx 18.8\, [m]
\]

### Step 2: Calculate the direction (bearing)
The angle \( \theta \) is the angle of the displacement vector relative to the negative x-axis (since the x-component is negative). The angle is calculated using the tangent function:

\[
\tan(\theta) = \frac{17.0}{|-8.00|}
\]

Solving for \( \theta \):

\[
\theta = \tan^{-1} \left( \frac{17.0}{8.00} \right)
\]

Now let's compute the angle.

\[
\theta \approx \tan^{-1}(2.125) \approx 64.8^\circ
\]

### Step 3: Determine the bearing
Since the x-component is negative and the y-component is positive, the displacement is towards the **North-West** direction.

Thus, the displacement vector is **18.8 m, 64.8° North of West**.

### Answer:
The correct answer is:
- \( \boxed{18.8 \, [m], 64.8^\circ \, \text{N of W}} \)

---

Would you like further details or have any questions? Here are some related questions:

1. How can displacement be described in terms of components?
2. What is the difference between scalar and vector quantities?
3. How do you determine the direction of a vector using its components?
4. What is the importance of the bearing in navigation or physics problems?
5. Can you convert between different coordinate systems, like polar and Cartesian?

**Tip:** Remember that the signs of the components are crucial for determining the correct quadrant and direction of a vector.

The displacement of a particle is Δr⃗ = −(8.00 [m]) î + (17.0 [m]) ĵ. What is the particle's displacement in magnitude-bearing form?

Learn how to calculate the displacement of a particle given in vector form. This problem involves the Pythagorean theorem to find the magnitude and trigonometry to determine the bearing angle. Suitable for high school students in grades 10-12.

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