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The problem involves two people pulling a box of weight 1400 N using ropes, with the forces given by vectors \((p \mathbf{i} + q \mathbf{j})\) N and \((p \mathbf{i} - q \mathbf{j})\) N, where \(\mathbf{i}\) and \(\mathbf{j}\) represent the unit vectors in the east and north directions, respectively. The friction force, 180 N, is acting westward, opposing the box's movement. The question asks to explain why the box moves in a straight line in the east direction.

### Part (a) – Why does the box move in a straight line in the east direction?
1. **Forces in the horizontal plane**: Since the forces are represented in terms of \(\mathbf{i}\) (east) and \(\mathbf{j}\) (north), the total force in the east and north directions will determine the direction of motion.
   
   The forces on the box due to the ropes are:
   - First force: \((p \mathbf{i} + q \mathbf{j})\)
   - Second force: \((p \mathbf{i} - q \mathbf{j})\)
   
2. **Resultant force in the north direction**: When adding these two forces, the north (or \(\mathbf{j}\)) components cancel out:
   \[
   q \mathbf{j} + (-q \mathbf{j}) = 0
   \]
   Therefore, the resultant force in the north direction is zero, meaning there is no movement in the north-south direction.

3. **Resultant force in the east direction**: The east (or \(\mathbf{i}\)) components add together:
   \[
   p \mathbf{i} + p \mathbf{i} = 2p \mathbf{i}
   \]
   The resultant force in the east direction is \(2p \mathbf{i}\), which means there is a net force pulling the box eastward.

4. **Friction force**: The friction force of 180 N acts in the west direction (opposite to \(\mathbf{i}\)). The box moves east if the eastward force \(2p \mathbf{i}\) exceeds the friction force.

Therefore, the box moves in a straight line in the east direction because the forces in the north-south direction cancel out, and the net force in the east direction exceeds the friction opposing it.

Would you like details on how to calculate \(p\) for specific values or a breakdown of the remaining parts of the question?

### 5 Related Questions:
1. How can the frictional force be calculated using the coefficient of friction and the normal force?
2. How would the direction of movement change if the forces were unequal in the north-south direction?
3. What would happen if the frictional force was greater than the net eastward force?
4. How do we calculate the magnitude of the resultant force from vector components?
5. Why is the box assumed to move in a straight line rather than a curve?

### Tip: 
When dealing with vectors in physics problems, it's useful to break them down into components along perpendicular axes (such as east-west and north-south) for easier calculations.

Two people are pulling a box of weight 1400 N using ropes. The forces in the ropes are given as (pi + qj) N and (pi - qj) N. The friction force of 180 N is directed west. Explain why the box moves in a straight line in the east direction.

Explore how two people pulling a box with forces represented as vectors results in eastward movement. Understand vector addition, friction, and Newton's laws through this physics problem.

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