To solve this problem, we need to understand how vector addition works and use the concept of the resultant vector. When adding two vectors, the magnitude of the resultant vector depends on the angle between the two vectors.

Given:

• Vector 1 magnitude (A) = 3 meters
• Vector 2 magnitude (B) = 4 meters

Resultant Vector Formula

The magnitude of the resultant vector $R$ formed by adding two vectors $A$ and $B$ at an angle $\theta$ is given by: $R = \sqrt{A^2 + B^2 + 2AB \cos \theta}$ Where:

• $A = 3$ m
• $B = 4$ m

(i) Magnitude of Resultant Vector $R = 7$ m

To achieve this, we use the formula: $7 = \sqrt{3^2 + 4^2 + 2 \times 3 \times 4 \cos \theta}$ Simplifying, we get: $49 = 9 + 16 + 24 \cos \theta$ $24 = 24 \cos \theta$ $\cos \theta = 1$ This implies that $\theta = 0^\circ$.

Conclusion: The two vectors should be added in the same direction (i.e., $\theta = 0^\circ$) to get a resultant vector of 7 meters.

(ii) Magnitude of Resultant Vector $R = 1$ m

Using the same formula: $1 = \sqrt{3^2 + 4^2 + 2 \times 3 \times 4 \cos \theta}$ Simplifying, we get: $1 = \sqrt{25 + 24 \cos \theta}$ $1 = \sqrt{25 + 24 \cos \theta}$ $1^2 = 25 + 24 \cos \theta$ $1 = 25 + 24 \cos \theta$ $24 \cos \theta = -24$ $\cos \theta = -1$ This implies that $\theta = 180^\circ$.

Conclusion: The two vectors should be added in opposite directions (i.e., $\theta = 180^\circ$) to get a resultant vector of 1 meter.

(iii) Magnitude of Resultant Vector $R = 5$ m

Using the same formula: $5 = \sqrt{3^2 + 4^2 + 2 \times 3 \times 4 \cos \theta}$ Simplifying, we get: $25 = 9 + 16 + 24 \cos \theta$ $25 = 25 + 24 \cos \theta$ $0 = 24 \cos \theta$ $\cos \theta = 0$ This implies that $\theta = 90^\circ$.

Conclusion: The two vectors should be added perpendicular to each other (i.e., $\theta = 90^\circ$) to get a resultant vector of 5 meters.

Summary:

• To get a resultant vector of 7 meters, the vectors should be added in the same direction (θ = 0°).
• To get a resultant vector of 1 meter, the vectors should be added in opposite directions (θ = 180°).
• To get a resultant vector of 5 meters, the vectors should be added perpendicularly (θ = 90°).

Do you want more details or have any questions?

Related Questions:

1. How does the angle between vectors affect the magnitude of their resultant?
2. What is the significance of the direction in vector addition?
3. How can the vector addition be graphically represented?
4. What happens if vectors are added at random angles?
5. Can the resultant of two vectors be less than the magnitude of either vector?

Tip: Understanding vector addition is crucial in physics, especially when dealing with forces, as they often act in different directions.