To add the vector pairs $\mathbf{A} = 50 \, \text{N}$ at $27^\circ$ and $\mathbf{B} = 70 \, \text{N}$ at $145^\circ$, we'll follow a systematic process similar to what was outlined before, breaking each vector into its horizontal and vertical components, summing those components, and then determining the magnitude and direction of the resultant vector.

Step 1: Decompose Each Vector into Components

Vector A:

$A_x = 50 \cos 27^\circ \approx 44.65 \, \text{N}$ $A_y = 50 \sin 27^\circ \approx 22.71 \, \text{N}$

Vector B:

$B_x = 70 \cos 145^\circ \approx -57.32 \, \text{N}$ $B_y = 70 \sin 145^\circ \approx 40.26 \, \text{N}$

Step 2: Sum the Components

Add the corresponding components of vectors $\mathbf{A}$ and $\mathbf{B}$:

Resultant $R_x$ (horizontal component):

$R_x = A_x + B_x = 44.65 + (-57.32) \approx -12.67 \, \text{N}$

Resultant $R_y$ (vertical component):

$R_y = A_y + B_y = 22.71 + 40.26 \approx 62.97 \, \text{N}$

Step 3: Calculate the Magnitude and Direction of the Resultant Vector

Magnitude of Resultant $R$:

$R = \sqrt{R_x^2 + R_y^2} = \sqrt{(-12.67)^2 + (62.97)^2} \approx 64.22 \, \text{N}$

Direction of Resultant $\theta_R$:

$\theta_R = \tan^{-1}\left(\frac{R_y}{R_x}\right) = \tan^{-1}\left(\frac{62.97}{-12.67}\right) \approx 101.4^\circ$

Final Answer:

• Magnitude of the Resultant Vector: $R \approx 64.22 \, \text{N}$
• Direction of the Resultant Vector: $\theta_R \approx 101.4^\circ$

This resultant vector has a magnitude of approximately 64.22 N and is directed at an angle of approximately 101.4° from the positive x-axis (standard counterclockwise rotation).

Do you have any further questions, or would you like more details?

Related Questions:

1. How does the resultant vector change if one of the vectors has a different magnitude or angle?
2. What happens to the resultant vector if we switch the angles of vectors $A$ and $B$?
3. Can you explain how to solve vector addition problems using the parallelogram method?
4. How can we determine the components if the vectors are given in unit vector notation?
5. What is the significance of the direction of the resultant vector in physical problems?

Tip: When working with angles, always ensure they are measured relative to the correct axis (usually from the positive x-axis).