The resultant of the two vectors `overset(rarr)a` and `overset(rarr)b` of magnitudes 6 and 10 respectively is at right angles to the vector `overset(rarr)a` Find their resultant and also the angle between the given vectors.

Options

A. `127^(@)`
B. `143^(@)`
C. `120^(@)`
D. `150^(@)`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the problem
We have two vectors, **A** and **B**, with magnitudes 6 and 10 respectively. The resultant vector **R** is at right angles to vector **A**. We need to find the magnitude of the resultant vector **R** and the angle between vectors **A** and **B**.

### Step 2: Set up the relationship using Pythagorean theorem
Since the resultant vector **R** is at right angles to vector **A**, we can use the Pythagorean theorem. The relationship can be expressed as:
\[ B^2 = A^2 + R^2 \]
Where:
- \( A = 6 \)
- \( B = 10 \)
- \( R \) is the resultant vector we need to find.

### Step 3: Substitute the known values
Substituting the known values into the equation:
\[ 10^2 = 6^2 + R^2 \]
This simplifies to:
\[ 100 = 36 + R^2 \]

### Step 4: Solve for R
Now, we can isolate \( R^2 \):
\[ R^2 = 100 - 36 \]
\[ R^2 = 64 \]
Taking the square root of both sides gives:
\[ R = \sqrt{64} = 8 \]
So, the magnitude of the resultant vector **R** is 8 units.

### Step 5: Find the angle between vectors A and B
Let the angle between vectors **A** and **B** be \( \theta \). Since we have a right triangle formed by vectors **A**, **B**, and the resultant **R**, we can use the tangent function to find \( \theta \):
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{A}{R} \]
Substituting the values:
\[ \tan(\theta) = \frac{6}{8} = \frac{3}{4} \]

### Step 6: Calculate the angle
To find \( \theta \), we take the arctangent:
\[ \theta = \tan^{-1}\left(\frac{3}{4}\right) \]
Calculating this gives:
\[ \theta \approx 36.87^\circ \text{ (or 37 degrees)} \]

### Step 7: Determine the total angle between vectors A and B
Since the resultant vector **R** is at right angles to vector **A**, the angle between vectors **A** and **B** is:
\[ \phi = 90^\circ + \theta = 90^\circ + 37^\circ = 127^\circ \]

### Final Results
- The magnitude of the resultant vector \( R \) is **8 units**.
- The angle between vectors **A** and **B** is **127 degrees**.