To solve the problem of finding the resultant vector of Vector A and Vector B, let's break down the vectors into their rectangular components (i.e., along the x-axis and y-axis).

We can use the following formulas to find the components of each vector:

• $A_x = A \cdot \cos(\theta_A)$

• $A_y = A \cdot \sin(\theta_A)$

• $B_x = B \cdot \cos(\theta_B)$

• $B_y = B \cdot \sin(\theta_B)$

Where:

• $A_x$ and $A_y$ are the x and y components of Vector A.
• $B_x$ and $B_y$ are the x and y components of Vector B.
• $\theta_A = -60^\circ$, $\theta_B = 30^\circ$.

Step 1: Calculate components of Vector A (magnitude = 8 N, direction = -60°)

$A_x = 8 \cdot \cos(-60^\circ) = 8 \cdot 0.5 = 4 \, N$ $A_y = 8 \cdot \sin(-60^\circ) = 8 \cdot (-0.866) = -6.928 \, N$

Step 2: Calculate components of Vector B (magnitude = 3 N, direction = 30°)

$B_x = 3 \cdot \cos(30^\circ) = 3 \cdot 0.866 = 2.598 \, N$ $B_y = 3 \cdot \sin(30^\circ) = 3 \cdot 0.5 = 1.5 \, N$

Step 3: Sum the components

Now, let's add the components of the two vectors together:

• $R_x = A_x + B_x = 4 + 2.598 = 6.598 \, N$
• $R_y = A_y + B_y = -6.928 + 1.5 = -5.428 \, N$

Step 4: Calculate the magnitude of the resultant vector

The magnitude $R$ of the resultant vector is given by: $R = \sqrt{R_x^2 + R_y^2} = \sqrt{(6.598)^2 + (-5.428)^2} = \sqrt{43.53 + 29.46} = \sqrt{72.99} \approx 8.54 \, N$

Step 5: Calculate the direction of the resultant vector

The direction $\theta_R$ of the resultant vector is given by: $\theta_R = \tan^{-1}\left(\frac{R_y}{R_x}\right) = \tan^{-1}\left(\frac{-5.428}{6.598}\right) \approx -39^\circ$

Thus, the resultant vector has a magnitude of approximately 8.54 N and a direction of about $-39^\circ$ (measured from the positive x-axis).

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Would you like more details or have any questions?

Related Questions:

1. How do we resolve vectors into components for 3D vectors?
2. Can we add vectors graphically and compare it with this method?
3. What happens if both vectors have the same angle but different magnitudes?
4. How would the problem change if Vector B had a negative angle?
5. How do we subtract vectors using components?

Tip:

Always make sure to use the correct sign for angles, especially when dealing with negative or clockwise directions.