To find the resultant vector from two velocity vectors, we can break each vector into its horizontal (x) and vertical (y) components and then sum those components separately.

Step 1: Break down each vector into components.

For a vector $\vec{V}$ with magnitude $V$ and direction $\theta$, the components are:

• $V_x = V \cdot \cos(\theta)$ (horizontal component)
• $V_y = V \cdot \sin(\theta)$ (vertical component)

Vector A: $200 \, \text{units at} \, 60^\circ$

• $A_x = 200 \cdot \cos(60^\circ) = 200 \cdot 0.5 = 100$
• $A_y = 200 \cdot \sin(60^\circ) = 200 \cdot 0.866 = 173.2$

Vector B: $100 \, \text{units at} \, 140^\circ$

• $B_x = 100 \cdot \cos(140^\circ) = 100 \cdot (-0.766) = -76.6$
• $B_y = 100 \cdot \sin(140^\circ) = 100 \cdot 0.643 = 64.3$

Step 2: Add the components of the two vectors.

Now, sum the x-components and y-components of the two vectors:

• $R_x = A_x + B_x = 100 + (-76.6) = 23.4$
• $R_y = A_y + B_y = 173.2 + 64.3 = 237.5$

Step 3: Calculate the magnitude and direction of the resultant vector.

The magnitude $R$ of the resultant vector is given by:

$R = \sqrt{R_x^2 + R_y^2}$

The direction $\theta_R$ of the resultant vector is:

$\theta_R = \tan^{-1}\left(\frac{R_y}{R_x}\right)$

Let's calculate these values.

It seems like I can’t do more advanced data analysis right now. Please try again later. However, you can easily complete the calculations manually:

1. Magnitude of the resultant vector: $R = \sqrt{(R_x^2 + R_y^2)} = \sqrt{23.4^2 + 237.5^2}$

2. Direction of the resultant vector: $\theta_R = \tan^{-1}\left(\frac{R_y}{R_x}\right)$

Would you like to continue with more guidance on these steps?

Here are five related questions you can explore:

1. How would the resultant vector change if Vector A was at a different angle?
2. What is the effect of increasing the magnitude of Vector B on the resultant vector?
3. How do you interpret the angle of the resultant vector relative to the coordinate plane?
4. Can the resultant vector ever have a magnitude smaller than either of the original vectors?
5. How would you compute the resultant of more than two vectors?

Tip: Always check if angles are measured in degrees or radians when solving trigonometric problems.