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

Answer

To find the angle between two vectors with magnitudes 3 units and 4 units, given different resultant magnitudes (1 unit, 5 units, and 7 units), we can use the formula for the magnitude of the resultant vector:

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

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.

### Step-by-Step Solution

**Given:**
- Magnitude of vector A, \( A = 3 \) units
- Magnitude of vector B, \( B = 4 \) units

#### (a) For \( R = 1 \) unit:

1. Substitute the values into the formula:
\[ 1 = \sqrt{3^2 + 4^2 + 2 \cdot 3 \cdot 4 \cos(\theta)} \]
2. Calculate \( 3^2 + 4^2 \):
\[ 3^2 + 4^2 = 9 + 16 = 25 \]
3. Now, the equation becomes:
\[ 1 = \sqrt{25 + 24 \cos(\theta)} \]
4. Square both sides:
\[ 1^2 = 25 + 24 \cos(\theta) \]
\[ 1 = 25 + 24 \cos(\theta) \]
5. Rearranging gives:
\[ 24 \cos(\theta) = 1 - 25 \]
\[ 24 \cos(\theta) = -24 \]
6. Therefore:
\[ \cos(\theta) = -1 \]
7. This implies:
\[ \theta = 180^\circ \]

#### (b) For \( R = 5 \) units:

1. Substitute the values into the formula:
\[ 5 = \sqrt{3^2 + 4^2 + 2 \cdot 3 \cdot 4 \cos(\theta)} \]
2. Using the previous calculation:
\[ 5 = \sqrt{25 + 24 \cos(\theta)} \]
3. Square both sides:
\[ 5^2 = 25 + 24 \cos(\theta) \]
\[ 25 = 25 + 24 \cos(\theta) \]
4. Rearranging gives:
\[ 24 \cos(\theta) = 25 - 25 \]
\[ 24 \cos(\theta) = 0 \]
5. Therefore:
\[ \cos(\theta) = 0 \]
6. This implies:
\[ \theta = 90^\circ \]

#### (c) For \( R = 7 \) units:

1. Substitute the values into the formula:
\[ 7 = \sqrt{3^2 + 4^2 + 2 \cdot 3 \cdot 4 \cos(\theta)} \]
2. Using the previous calculation:
\[ 7 = \sqrt{25 + 24 \cos(\theta)} \]
3. Square both sides:
\[ 7^2 = 25 + 24 \cos(\theta) \]
\[ 49 = 25 + 24 \cos(\theta) \]
4. Rearranging gives:
\[ 24 \cos(\theta) = 49 - 25 \]
\[ 24 \cos(\theta) = 24 \]
5. Therefore:
\[ \cos(\theta) = 1 \]
6. This implies:
\[ \theta = 0^\circ \]

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