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Determine:

a. Las componentes rectangualares de los vectores 1 y 2.
b. Exprese los vectores 1 y 2 en terminos de los vectores unitarios.
c. A partir del punto b, sume los vectores para encontrar el vector resultante.
d. Calcule la magnitud y direcciòn del vector resultante.

To solve the given problem, let's break it down step by step.

### Given Information:
- **Vector V1:**
  - Magnitude \( V_1 = 80.0 \, \text{units} \)
  - Angle \( \theta_1 = 60.0^\circ \) from the positive x-axis.

- **Vector V2:**
  - Magnitude \( V_2 = 120.0 \, \text{units} \)
  - Angle \( \theta_2 = 75.0^\circ \) from the negative x-axis.

### Steps:

#### **a. Calculate the rectangular components of the vectors:**

1. **For \( V_1 \):**
   \[
   V_{1x} = V_1 \cos(\theta_1) = 80.0 \cos(60.0^\circ)
   \]
   \[
   V_{1y} = V_1 \sin(\theta_1) = 80.0 \sin(60.0^\circ)
   \]

2. **For \( V_2 \):**
   Since \( V_2 \) is at an angle of \( 75.0^\circ \) with the negative x-axis, the angle with the positive x-axis is \( 180^\circ - 75^\circ = 105^\circ \).
   \[
   V_{2x} = V_2 \cos(105.0^\circ) = 120.0 \cos(105.0^\circ)
   \]
   \[
   V_{2y} = V_2 \sin(105.0^\circ) = 120.0 \sin(105.0^\circ)
   \]

Let's calculate these components.

#### **b. Express the vectors in terms of unit vectors:**

1. \( \mathbf{V_1} = V_{1x} \mathbf{i} + V_{1y} \mathbf{j} \)
2. \( \mathbf{V_2} = V_{2x} \mathbf{i} + V_{2y} \mathbf{j} \)

#### **c. Sum the vectors to find the resultant vector:**

\[
\mathbf{V_R} = \mathbf{V_1} + \mathbf{V_2}
\]
\[
V_{Rx} = V_{1x} + V_{2x}, \quad V_{Ry} = V_{1y} + V_{2y}
\]

#### **d. Calculate the magnitude and direction of the resultant vector:**

1. **Magnitude:**
   \[
   |\mathbf{V_R}| = \sqrt{V_{Rx}^2 + V_{Ry}^2}
   \]

2. **Direction:**
   \[
   \theta_R = \tan^{-1}\left(\frac{V_{Ry}}{V_{Rx}}\right)
   \]

Now, I'll proceed to calculate these values.### Solution:

#### **a. Rectangular components of the vectors:**
1. **Vector \( V_1 \):**
   - \( V_{1x} = 40.0 \, \text{units} \)
   - \( V_{1y} = 69.28 \, \text{units} \)

2. **Vector \( V_2 \):**
   - \( V_{2x} = -31.06 \, \text{units} \)
   - \( V_{2y} = 115.91 \, \text{units} \)

#### **b. Vectors in terms of unit vectors:**
1. \( \mathbf{V_1} = 40.0 \mathbf{i} + 69.28 \mathbf{j} \)
2. \( \mathbf{V_2} = -31.06 \mathbf{i} + 115.91 \mathbf{j} \)

#### **c. Resultant vector components:**
1. \( V_{Rx} = 8.94 \, \text{units} \)
2. \( V_{Ry} = 185.19 \, \text{units} \)

Thus, \( \mathbf{V_R} = 8.94 \mathbf{i} + 185.19 \mathbf{j} \).

#### **d. Magnitude and direction of the resultant vector:**
1. **Magnitude:** \( |\mathbf{V_R}| = 185.41 \, \text{units} \)
2. **Direction:** \( \theta_R = 87.24^\circ \) from the positive x-axis.

Would you like more details on any of these calculations, or do you have any questions?

### Related Questions:
1. How do you determine the quadrant in which the resultant vector lies?
2. What are the effects of changing the angles of vectors on the resultant vector?
3. How can the components method be applied to 3D vectors?
4. What is the significance of using unit vectors in vector representation?
5. How would you find the angle between two given vectors?

### Tip:
Always check the sign of your components when summing vectors, as this determines the correct direction of the resultant vector.

Learn how to solve vector addition and magnitude calculation problems step-by-step with examples. This problem covers calculating rectangular components of vectors, expressing vectors in terms of unit vectors, summing vectors, and finding the magnitude and direction of the resultant vector.

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