Two vectors have magnitudes 3 unit and 4 unit respectively. What should be the angel between them if the magnitude of the resultant ils a. 1 unit, b. 5 unit and c. 7 unit.

Explanation

To solve the problem, we will use the formula for the magnitude of the resultant vector when two vectors are involved. The formula is:

\[ R = \sqrt{A^2 + B^2 + 2AB \cos \theta} \]

where:
- \( R \) is the magnitude of the resultant vector,
- \( A \) and \( B \) are the magnitudes of the two vectors,
- \( \theta \) is the angle between the two vectors.

Given:
- \( A = 3 \) units,
- \( B = 4 \) units.

We will find the angle \( \theta \) for three cases of the resultant vector \( R \): 1 unit, 5 units, and 7 units.

### Part (a): Resultant \( R = 1 \) unit

1. Substitute the values into the formula:
\[ 1 = \sqrt{3^2 + 4^2 + 2 \cdot 3 \cdot 4 \cos \theta} \]
This simplifies to:
\[ 1 = \sqrt{9 + 16 + 24 \cos \theta} \]
\[ 1 = \sqrt{25 + 24 \cos \theta} \]

2. Square both sides:
\[ 1^2 = 25 + 24 \cos \theta \]
\[ 1 = 25 + 24 \cos \theta \]

3. Rearranging gives:
\[ 24 \cos \theta = 1 - 25 \]
\[ 24 \cos \theta = -24 \]

4. Divide by 24:
\[ \cos \theta = -1 \]

5. Find \( \theta \):
\[ \theta = \cos^{-1}(-1) = 180^\circ \]

### Part (b): Resultant \( R = 5 \) units

1. Substitute the values into the formula:
\[ 5 = \sqrt{3^2 + 4^2 + 2 \cdot 3 \cdot 4 \cos \theta} \]
This simplifies to:
\[ 5 = \sqrt{25 + 24 \cos \theta} \]

2. Square both sides:
\[ 5^2 = 25 + 24 \cos \theta \]
\[ 25 = 25 + 24 \cos \theta \]

3. Rearranging gives:
\[ 24 \cos \theta = 25 - 25 \]
\[ 24 \cos \theta = 0 \]

4. Divide by 24:
\[ \cos \theta = 0 \]

5. Find \( \theta \):
\[ \theta = \cos^{-1}(0) = 90^\circ \]

### Part (c): Resultant \( R = 7 \) units

1. Substitute the values into the formula:
\[ 7 = \sqrt{3^2 + 4^2 + 2 \cdot 3 \cdot 4 \cos \theta} \]
This simplifies to:
\[ 7 = \sqrt{25 + 24 \cos \theta} \]

2. Square both sides:
\[ 7^2 = 25 + 24 \cos \theta \]
\[ 49 = 25 + 24 \cos \theta \]

3. Rearranging gives:
\[ 24 \cos \theta = 49 - 25 \]
\[ 24 \cos \theta = 24 \]

4. Divide by 24:
\[ \cos \theta = 1 \]

5. Find \( \theta \):
\[ \theta = \cos^{-1}(1) = 0^\circ \]

### Summary of Results:
- For \( R = 1 \) unit, \( \theta = 180^\circ \)
- For \( R = 5 \) units, \( \theta = 90^\circ \)
- For \( R = 7 \) units, \( \theta = 0^\circ \)