To approach this problem, let's visualize how two displacement vectors can combine to yield different resultant magnitudes. The two displacements have magnitudes of 1.8 m and 2.4 m. These vectors can be arranged in different ways to produce varying resultant magnitudes, depending on their direction.

(a) Resultant of 4.2 m

• Explanation: A resultant of 4.2 m is the maximum possible resultant, which occurs when the two displacements are in the same direction. The two vectors will add directly as $1.8 \, \text{m} + 2.4 \, \text{m} = 4.2 \, \text{m}$.
• Sketch: Draw two vectors pointing in the same direction, with one vector having a length of 1.8 m and the other 2.4 m. The combined vector will measure 4.2 m.

(b) Resultant of 0.6 m

• Explanation: This is the minimum possible resultant, which occurs when the two displacements are in exact opposite directions. In this case, the magnitude of the resultant is the difference of the two displacements: $|1.8 \, \text{m} - 2.4 \, \text{m}| = 0.6 \, \text{m}$.
• Sketch: Draw two vectors pointing in opposite directions, with one vector measuring 1.8 m and the other 2.4 m. The resultant vector will measure 0.6 m.

(c) Resultant of 3.0 m

• Explanation: To get a resultant of 3.0 m, the two vectors must be at some angle to each other. The general formula for the resultant of two vectors at an angle $\theta$ is given by: $R = \sqrt{(1.8)^2 + (2.4)^2 + 2 \times 1.8 \times 2.4 \times \cos(\theta)}$ Solving for $\theta$ using the condition $R = 3.0 \, \text{m}$, we can find the required angle.
• Sketch: Draw two vectors at an angle (approximately 60-90 degrees apart) where one vector is 1.8 m and the other is 2.4 m. The resultant vector will be about 3.0 m.

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Would you like a detailed breakdown of the angles for case (c), or any further explanation?

Here are 5 follow-up questions for deeper understanding:

1. What is the vector addition formula for two vectors at an angle?
2. How would the magnitude of the resultant change if the angle between the vectors increased?
3. What would happen if both vectors pointed in the same direction but one was longer?
4. How can vector addition be applied to physical scenarios like forces or velocities?
5. Why does adding vectors in opposite directions yield the minimum resultant?

Tip: When vectors are perpendicular, the resultant is calculated using the Pythagorean theorem.