Resultant Vector by Rectangular Components

Problem

Find the resultant vector for part (e).

Step 1: Decompose the Vectors

To find the resultant vector, decompose each vector into its $x$ and $y$ components using

$$V_x = V\cos(\theta)$$

and

$$V_y = V\sin(\theta).$$

Step 2: Sum the Components

After calculating the components for all four vectors, sum the $x$-components and $y$-components separately:

$$R_x = 12.8 + 7.2 - 20 + 10 = 10\text{ N}$$

and

$$R_y = 9.6 - 9.6 + 0 + 0 = 0\text{ N}.$$

Step 3: Determine the Resultant

Since the $y$-component is zero, the resultant vector lies entirely along the $X$-axis.

Therefore, the final resultant is

$$\boxed{10\text{ N}}$$

directed along the $X$-axis.