The resultant of two vectors of magnitudes `3` units and `5` units is perpendicular to `3` units.The angle between the vectors is

Options

A. `127^(@)`
B. `120^(@)`
C. `75^(@)`
D. `150^(@)`

Explanation

To solve the problem of finding the angle between two vectors of magnitudes 3 units and 5 units, where the resultant is perpendicular to the vector of 3 units, we can follow these steps:

### Step 1: Understand the Problem
We have two vectors, \( \vec{A} \) with a magnitude of 3 units and \( \vec{B} \) with a magnitude of 5 units. The resultant vector \( \vec{R} \) is perpendicular to \( \vec{A} \).

### Step 2: Apply the Pythagorean Theorem
Since \( \vec{R} \) is perpendicular to \( \vec{A} \), we can use the Pythagorean theorem to find the magnitude of the resultant vector \( \vec{R} \):
\[ R^2 = A^2 + B^2 \]
Here, \( A = 3 \) units and \( B = 5 \) units. However, since \( \vec{R} \) is perpendicular to \( \vec{A} \), we can rearrange this to:
\[ R^2 = B^2 - A^2 \]
Substituting the values:
\[ R^2 = 5^2 - 3^2 = 25 - 9 = 16 \]
Thus,
\[ R = \sqrt{16} = 4 \text{ units} \]

### Step 3: Use the Cosine Rule to Find the Angle
Now, we need to find the angle \( \theta \) between the two vectors. We can use the cosine rule:
\[ R^2 = A^2 + B^2 - 2AB \cos(\theta) \]
Substituting the known values:
\[ 4^2 = 3^2 + 5^2 - 2 \times 3 \times 5 \cos(\theta) \]
This simplifies to:
\[ 16 = 9 + 25 - 30 \cos(\theta) \]
\[ 16 = 34 - 30 \cos(\theta) \]
Rearranging gives:
\[ 30 \cos(\theta) = 34 - 16 \]
\[ 30 \cos(\theta) = 18 \]
\[ \cos(\theta) = \frac{18}{30} = \frac{3}{5} \]

### Step 4: Calculate the Angle
Now, we find \( \theta \):
\[ \theta = \cos^{-1}\left(\frac{3}{5}\right) \]
Using a calculator or trigonometric tables, we find:
\[ \theta \approx 53^\circ \]

### Step 5: Find the Angle Between the Vectors
Since the angle between the two vectors is \( \theta \), we can find the angle \( \theta' \) between the resultant and the vector \( \vec{B} \):
\[ \theta' = 180^\circ - \theta \]
Thus,
\[ \theta' = 180^\circ - 53^\circ = 127^\circ \]

### Final Answer
The angle between the two vectors is approximately \( 127^\circ \).

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