Combine the following two velocity vectors to find the resultant vector. Vector
A = 200 $ % at 60° and Vector B = 100 $ % at 140° .

To combine the two vectors, we need to break each vector into its horizontal (x) and vertical (y) components and then sum the components.

### Step 1: Decompose each vector into components.

#### For Vector A (200 m/s at 60°):
- The horizontal component (Ax) is given by $ A_x = A \cdot \cos(\theta_A) $.
- The vertical component (Ay) is given by $ A_y = A \cdot \sin(\theta_A) $.

\[
A_x = 200 \cdot \cos(60^\circ) = 200 \cdot 0.5 = 100 \, \text{m/s}
\]
\[
A_y = 200 \cdot \sin(60^\circ) = 200 \cdot 0.866 = 173.2 \, \text{m/s}
\]

#### For Vector B (100 m/s at 140°):
- The horizontal component (Bx) is given by $ B_x = B \cdot \cos(\theta_B) $.
- The vertical component (By) is given by $ B_y = B \cdot \sin(\theta_B) $.

\[
B_x = 100 \cdot \cos(140^\circ) = 100 \cdot (-0.766) = -76.6 \, \text{m/s}
\]
\[
B_y = 100 \cdot \sin(140^\circ) = 100 \cdot 0.643 = 64.3 \, \text{m/s}
\]

### Step 2: Sum the components.
- Resultant horizontal component (Rx): $ R_x = A_x + B_x $
- Resultant vertical component (Ry): $ R_y = A_y + B_y $

\[
R_x = 100 + (-76.6) = 23.4 \, \text{m/s}
\]
\[
R_y = 173.2 + 64.3 = 237.5 \, \text{m/s}
\]

### Step 3: Find the magnitude of the resultant vector.
The magnitude $ R $ is given by the Pythagorean theorem:
\[
R = \sqrt{R_x^2 + R_y^2}
\]
\[
R = \sqrt{(23.4)^2 + (237.5)^2} = \sqrt{547.56 + 56412.25} = \sqrt{56959.81} \approx 238.7 \, \text{m/s}
\]

### Step 4: Find the direction of the resultant vector.
The direction (angle $ \theta_R $) of the resultant vector is given by:
\[
\theta_R = \tan^{-1} \left( \frac{R_y}{R_x} \right)
\]
\[
\theta_R = \tan^{-1} \left( \frac{237.5}{23.4} \right) \approx \tan^{-1}(10.15) \approx 84.3^\circ
\]

### Final Result:
The magnitude of the resultant vector is approximately **238.7 m/s**, and its direction is **84.3°**.

Would you like more details or have any questions?

Here are some related questions:
1. How do you convert between polar and Cartesian coordinates for vectors?
2. What is the difference between vector addition and scalar addition?
3. How do you calculate the angle between two vectors?
4. How would the problem change if Vector B were directed in the negative direction?
5. How does the law of cosines apply to finding the magnitude of the resultant?

**Tip:** Always make sure the angles you use in trigonometric functions are consistent with your calculator's mode (degrees or radians).

Combine the following two velocity vectors to find the resultant vector. Vector A = 200 m/s at 60° and Vector B = 100 m/s at 140°.

Learn how to combine two velocity vectors, 200 m/s at 60° and 100 m/s at 140°, using vector addition and trigonometry. Follow a step-by-step solution to break the vectors into components, sum them, and find the resultant vector with a magnitude of approximately 238.7 m/s and direction of 84.3°. Suitable for Grades 10-12.

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