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Given the displacements shown in the figure, determine the direction of the resultant with respect to the positive “x” axis and counterclockwise. Consider a = 596 m and b = 442 m.



To determine the direction of the resultant displacement with respect to the positive x-axis (counterclockwise), we will use vector addition.

### Step 1: Break Down the Displacements into Components

Given:
- \( a = 596 \, \text{m} \) 
- \( b = 442 \, \text{m} \)
- The first displacement \( a \) is directed at an angle of \( 30^\circ \) above the x-axis.
- The second displacement \( b \) is directed at an angle of \( 45^\circ \) below the negative x-axis.

#### Components of Displacement \( a \):
- \( a_x = a \cos(30^\circ) \)
- \( a_y = a \sin(30^\circ) \)

#### Components of Displacement \( b \):
- \( b_x = -b \cos(45^\circ) \) (negative because it's along the negative x-axis)
- \( b_y = -b \sin(45^\circ) \) (negative because it's below the x-axis)

### Step 2: Calculate Components
Let's calculate the numerical values:

- \( a_x = 596 \times \cos(30^\circ) \)
- \( a_y = 596 \times \sin(30^\circ) \)
- \( b_x = -442 \times \cos(45^\circ) \)
- \( b_y = -442 \times \sin(45^\circ) \)

### Step 3: Find the Resultant Components
- \( R_x = a_x + b_x \)
- \( R_y = a_y + b_y \)

### Step 4: Direction of the Resultant Displacement
- Direction: \( \theta_R = \tan^{-1}\left(\frac{R_y}{R_x}\right) \)

I'll calculate these steps now.The resultant displacement has the following components:
- \( R_x \approx 203.61 \, \text{m} \)
- \( R_y \approx -14.54 \, \text{m} \)

The direction of the resultant displacement with respect to the positive x-axis (measured counterclockwise) is approximately **\(-4.08^\circ\)**. This indicates that the resultant vector is slightly below the positive x-axis.

Do you want more details or have any questions?

Here are some related questions you might find helpful:
1. How do the x and y components of vectors affect the resultant direction?
2. Why is the angle of the resultant displacement negative in this case?
3. How would the resultant displacement change if one of the vectors was reversed?
4. What is the significance of the angle being close to the x-axis?
5. How can you determine the magnitude of the resultant displacement?

**Tip:** When dealing with angles, always pay attention to the reference direction (like counterclockwise from the positive x-axis) to avoid confusion about the vector's actual orientation.

Learn how to calculate the direction of the resultant displacement using vector addition with given displacements. Detailed step-by-step solution includes trigonometric calculations and vector components. Suitable for high school students studying vectors and trigonometry.

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