Vector A has a magnitude of 8.0 at an angle of 60º from the x-axis. Vector B has a magnitude of 6.0 at an angle of -30º from the x-axis. Determine the resultant vector.

To determine the resultant vector \( \vec{R} = \vec{A} + \vec{B} \), we will break each vector into its components, add them, and then find the magnitude and direction of the resultant vector.

### Step 1: Calculate the components of vector \( \vec{A} \).
- Magnitude \( |\vec{A}| = 8.0 \)
- Angle \( \theta_A = 60^\circ \)

The x and y components of \( \vec{A} \) are:
\[
A_x = |\vec{A}| \cos(\theta_A) = 8.0 \cos(60^\circ)
\]
\[
A_y = |\vec{A}| \sin(\theta_A) = 8.0 \sin(60^\circ)
\]

Substituting the trigonometric values:
\[
A_x = 8.0 \times 0.5 = 4.0
\]
\[
A_y = 8.0 \times 0.866 \approx 6.928
\]

So, the components of vector \( \vec{A} \) are:
\[
\vec{A} = (4.0, 6.928)
\]

### Step 2: Calculate the components of vector \( \vec{B} \).
- Magnitude \( |\vec{B}| = 6.0 \)
- Angle \( \theta_B = -30^\circ \)

The x and y components of \( \vec{B} \) are:
\[
B_x = |\vec{B}| \cos(\theta_B) = 6.0 \cos(-30^\circ)
\]
\[
B_y = |\vec{B}| \sin(\theta_B) = 6.0 \sin(-30^\circ)
\]

Substituting the trigonometric values:
\[
B_x = 6.0 \times 0.866 \approx 5.196
\]
\[
B_y = 6.0 \times (-0.5) = -3.0
\]

So, the components of vector \( \vec{B} \) are:
\[
\vec{B} = (5.196, -3.0)
\]

### Step 3: Add the components to find the resultant vector \( \vec{R} \).
\[
R_x = A_x + B_x = 4.0 + 5.196 = 9.196
\]
\[
R_y = A_y + B_y = 6.928 + (-3.0) = 3.928
\]

### Step 4: Determine the magnitude and direction of the resultant vector.
The magnitude \( |\vec{R}| \) is:
\[
|\vec{R}| = \sqrt{R_x^2 + R_y^2} = \sqrt{(9.196)^2 + (3.928)^2}
\]
\[
|\vec{R}| \approx \sqrt{84.57 + 15.43} = \sqrt{100.0} = 10.0
\]

The direction \( \theta_R \) relative to the x-axis is:
\[
\theta_R = \tan^{-1}\left(\frac{R_y}{R_x}\right) = \tan^{-1}\left(\frac{3.928}{9.196}\right)
\]
\[
\theta_R \approx \tan^{-1}(0.427) \approx 23.0^\circ
\]

### Final Result:
The resultant vector \( \vec{R} \) has:
- **Magnitude:** \( 10.0 \)
- **Direction:** \( 23.0^\circ \) from the x-axis

Would you like to explore more details or ask any questions?

### Follow-up Questions:
1. How does the angle of a vector affect its x and y components?
2. What is the importance of the Pythagorean theorem in finding vector magnitudes?
3. Can vectors be added in polar form instead of component form?
4. How does the direction of the resultant vector change if the angle of one vector changes?
5. How do you subtract vectors using components?
6. What real-world applications require vector addition?
7. How do you convert a vector from polar to Cartesian coordinates?
8. What is a unit vector, and how is it calculated?

**Tip:** Always double-check the angle direction (positive vs. negative) as it significantly impacts the vector's components.

Learn how to calculate the resultant vector from two vectors, A and B, given their magnitudes and angles relative to the x-axis. This problem involves vector addition using trigonometric components, determining magnitude, and direction. Suitable for students in Grades 10-12.

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