Vector A = 50 N at 27 degrees, Vector B = 70 N at 145 degrees

To solve the problem with vectors \( A \) and \( B \), we will break each vector into its components and then find the resultant vector by adding these components.

### Given Data:
- **Vector A**: \( A = 50 \, \text{N} \) at \( 27^\circ \)
- **Vector B**: \( B = 70 \, \text{N} \) at \( 145^\circ \)

### Step 1: Break Each Vector into Components

The components of a vector can be found using the following formulas:
\[
A_x = A \cos \theta_A \quad \text{and} \quad A_y = A \sin \theta_A
\]
\[
B_x = B \cos \theta_B \quad \text{and} \quad B_y = B \sin \theta_B
\]

#### Vector A:
\[
A_x = 50 \cos 27^\circ
\]
\[
A_y = 50 \sin 27^\circ
\]

#### Vector B:
\[
B_x = 70 \cos 145^\circ
\]
\[
B_y = 70 \sin 145^\circ
\]

### Step 2: Calculate the Components

I'll calculate these components for you.

#### Vector A Components:
\[
A_x = 50 \cos 27^\circ \approx 44.65 \, \text{N}
\]
\[
A_y = 50 \sin 27^\circ \approx 22.71 \, \text{N}
\]

#### Vector B Components:
\[
B_x = 70 \cos 145^\circ \approx -57.32 \, \text{N}
\]
\[
B_y = 70 \sin 145^\circ \approx 40.26 \, \text{N}
\]

### Step 3: Find the Resultant Components

Add the components of vectors \( A \) and \( B \):
\[
R_x = A_x + B_x \approx 44.65 - 57.32 \approx -12.67 \, \text{N}
\]
\[
R_y = A_y + B_y \approx 22.71 + 40.26 \approx 62.97 \, \text{N}
\]

### Step 4: Calculate the Magnitude and Direction of the Resultant Vector

The magnitude \( R \) of the resultant vector is given by:
\[
R = \sqrt{R_x^2 + R_y^2}
\]
\[
R = \sqrt{(-12.67)^2 + (62.97)^2} \approx 64.22 \, \text{N}
\]

The direction \( \theta_R \) of the resultant vector is:
\[
\theta_R = \tan^{-1}\left(\frac{R_y}{R_x}\right)
\]
\[
\theta_R = \tan^{-1}\left(\frac{62.97}{-12.67}\right) \approx 101.4^\circ
\]

### Final Answer:
- **Magnitude of the Resultant Vector**: \( R \approx 64.22 \, \text{N} \)
- **Direction of the Resultant Vector**: \( \theta_R \approx 101.4^\circ \)

Would you like more details or have any other questions?

### Related Questions:
1. How do we add more than two vectors using components?
2. What if one of the vectors had a negative magnitude?
3. How can we solve vector problems using the graphical method?
4. How would these vectors affect each other if they were forces acting on an object?
5. Can vectors be represented in three-dimensional space, and how would that change the calculations?

**Tip:** Always ensure that the angle measurements match the coordinate system being used (e.g., standard position angles in physics).

Learn how to solve vector problems with detailed step-by-step solutions. This problem covers vectors A = 50 N at 27 degrees and B = 70 N at 145 degrees, breaking each into components and finding the resultant vector's magnitude and direction.

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