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The problem in the image involves adding four displacement vectors, labeled \( A \), \( B \), \( C \), and \( D \), and finding the magnitude and direction of the resultant vector. The angles and magnitudes of the vectors are given as follows:

- \( A = 16.0 \, \text{m} \) at \( 35^\circ \) above the positive \( x \)-axis.
- \( B = 10.0 \, \text{m} \) along the positive \( y \)-axis (\( 90^\circ \)).
- \( C = 11.0 \, \text{m} \) at \( 35^\circ \) below the negative \( x \)-axis.
- \( D = 29.0 \, \text{m} \) at \( 50^\circ \) below the positive \( x \)-axis.

### Solution Steps:
To find the resultant vector, we need to calculate the sum of the \( x \)-components and \( y \)-components of each vector. Then, we use the Pythagorean theorem to find the magnitude of the resultant vector, and the inverse tangent to find its direction.

#### 1. Resolve the vectors into components:
- \( A_x = A \cos(35^\circ) = 16.0 \cos(35^\circ) \)
- \( A_y = A \sin(35^\circ) = 16.0 \sin(35^\circ) \)

- \( B_x = 0 \) (since \( B \) is along the \( y \)-axis)
- \( B_y = B = 10.0 \, \text{m} \)

- \( C_x = C \cos(35^\circ) = -11.0 \cos(35^\circ) \) (negative since it's to the left of the origin)
- \( C_y = C \sin(35^\circ) = -11.0 \sin(35^\circ) \) (negative since it's downward)

- \( D_x = D \cos(50^\circ) = 29.0 \cos(50^\circ) \)
- \( D_y = D \sin(50^\circ) = -29.0 \sin(50^\circ) \) (negative since it's downward)

#### 2. Sum the components:
- \( R_x = A_x + B_x + C_x + D_x \)
- \( R_y = A_y + B_y + C_y + D_y \)

#### 3. Find the magnitude of the resultant vector:
\[ R = \sqrt{R_x^2 + R_y^2} \]

#### 4. Find the direction of the resultant vector:
\[ \theta = \tan^{-1} \left( \frac{R_y}{R_x} \right) \]

I will now calculate these values.The magnitude and direction of the resultant vector are as follows:

- **Magnitude**: \( 24.58 \, \text{m} \)
- **Direction**: \( -22.35^\circ \) (measured counterclockwise from the positive \( x \)-axis, so this is 22.35° below the positive \( x \)-axis).

Would you like further clarification or details on any of the steps? Here are five related questions you can explore:

1. How do we add vectors graphically versus analytically?
2. How would the results change if one of the vectors had a different direction?
3. What is the significance of breaking vectors into components?
4. How does the resultant vector help in understanding physical motion?
5. How can we interpret negative angles in vector direction?

**Tip**: Always break vectors into components to simplify problems involving vector addition!

Determine the (a) magnitude and (b) direction for the resultant that occurs when the vectors A = 16.0 m, B = 10.0 m, C = 11.0 m, and D = 29.0 m are added together. Specify the direction as a positive (counterclockwise) angle from the +x axis.

Learn how to find the magnitude and direction of the resultant vector when adding four displacement vectors: A = 16.0 m, B = 10.0 m, C = 11.0 m, and D = 29.0 m. Includes vector components, the Pythagorean theorem, and trigonometric ratios. Ideal for Grades 11-12 physics students.

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