What is the resultant of the following 3 concurrent vectors: 600 lbs North, 300 lbs East, and 200 lbs South?

To find the resultant of the three concurrent vectors, we can break them down into their components along the north-south (vertical) and east-west (horizontal) axes. Here’s how you can do it:

### Step 1: Determine the components along each axis.
- **North (600 lbs):** This is entirely in the vertical direction (positive y-axis).
  \[
  \text{Vertical Component (North)} = +600 \, \text{lbs}
  \]
- **East (300 lbs):** This is entirely in the horizontal direction (positive x-axis).
  \[
  \text{Horizontal Component (East)} = +300 \, \text{lbs}
  \]
- **South (200 lbs):** This is entirely in the vertical direction but opposite to North (negative y-axis).
  \[
  \text{Vertical Component (South)} = -200 \, \text{lbs}
  \]

### Step 2: Sum the components along each axis.
- **Resultant Vertical Component:**
  \[
  \text{Resultant Vertical} = +600 \, \text{lbs} - 200 \, \text{lbs} = +400 \, \text{lbs}
  \]
- **Resultant Horizontal Component:**
  \[
  \text{Resultant Horizontal} = +300 \, \text{lbs}
  \]

### Step 3: Calculate the magnitude of the resultant vector.
The magnitude of the resultant vector \( R \) can be found using the Pythagorean theorem:
\[
R = \sqrt{(\text{Resultant Horizontal})^2 + (\text{Resultant Vertical})^2}
\]
\[
R = \sqrt{(300)^2 + (400)^2}
\]
\[
R = \sqrt{90000 + 160000}
\]
\[
R = \sqrt{250000} = 500 \, \text{lbs}
\]

### Step 4: Determine the direction of the resultant vector.
The direction \( \theta \) of the resultant vector with respect to the east direction can be found using the arctangent function:
\[
\theta = \tan^{-1}\left(\frac{\text{Resultant Vertical}}{\text{Resultant Horizontal}}\right)
\]
\[
\theta = \tan^{-1}\left(\frac{400}{300}\right) = \tan^{-1}\left(\frac{4}{3}\right)
\]
\[
\theta \approx 53.13^\circ \text{ North of East}
\]

### Final Answer:
The resultant of the three concurrent vectors is **500 lbs** at an angle of **53.13° North of East**.

Would you like more details, or do you have any questions?

Here are 5 related questions:
1. How would the resultant change if the southward vector was 400 lbs instead of 200 lbs?
2. What is the importance of the direction when finding the resultant of multiple vectors?
3. How can you represent the resultant vector graphically?
4. What happens if all the vectors are in the same direction?
5. How would you calculate the resultant if the vectors were not at right angles?

**Tip:** When dealing with vector problems, always break them into components and use trigonometry for accurate results.

Find the resultant of the vectors 600 lbs North, 300 lbs East, and 200 lbs South using vector addition. This detailed solution covers vector components, the Pythagorean theorem, and trigonometric functions to calculate both the magnitude and direction of the resultant. Suitable for students in Grades 9-12.

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