What is the vector equally inclined to the vectors `hat(i)+ 3hat(j) and 3 hat(i) + hat(j)` ?

Options

A. a) hat(i) + hat(j)`
B. b) 2 hat(i)- hat(j)`
C. c) 2 hat(i) + hat(j)`
D. d) none of these

Answer

To find a vector that is equally inclined to the vectors \( \hat{i} + 3\hat{j} \) and \( 3\hat{i} + \hat{j} \), we can follow these steps:

### Step 1: Define the Vectors
Let:
- \( \mathbf{A} = \hat{i} + 3\hat{j} \)
- \( \mathbf{B} = 3\hat{i} + \hat{j} \)

### Step 2: Find the Unit Vectors
First, we need to find the unit vectors in the direction of \( \mathbf{A} \) and \( \mathbf{B} \).

The magnitude of \( \mathbf{A} \):
\[ |\mathbf{A}| = \sqrt{(1)^2 + (3)^2} = \sqrt{1 + 9} = \sqrt{10} \]
The unit vector in the direction of \( \mathbf{A} \):
\[ \hat{A} = \frac{\mathbf{A}}{|\mathbf{A}|} = \frac{\hat{i} + 3\hat{j}}{\sqrt{10}} = \frac{1}{\sqrt{10}}\hat{i} + \frac{3}{\sqrt{10}}\hat{j} \]

The magnitude of \( \mathbf{B} \):
\[ |\mathbf{B}| = \sqrt{(3)^2 + (1)^2} = \sqrt{9 + 1} = \sqrt{10} \]
The unit vector in the direction of \( \mathbf{B} \):
\[ \hat{B} = \frac{\mathbf{B}}{|\mathbf{B}|} = \frac{3\hat{i} + \hat{j}}{\sqrt{10}} = \frac{3}{\sqrt{10}}\hat{i} + \frac{1}{\sqrt{10}}\hat{j} \]

### Step 3: Find the Vector Equally Inclined
A vector that is equally inclined to both \( \mathbf{A} \) and \( \mathbf{B} \) can be expressed as a linear combination of \( \hat{A} \) and \( \hat{B} \):
\[ \mathbf{C} = k(\hat{A} + \hat{B}) \]
where \( k \) is a scalar.

Calculating \( \hat{A} + \hat{B} \):
\[ \hat{A} + \hat{B} = \left(\frac{1}{\sqrt{10}} + \frac{3}{\sqrt{10}}\right)\hat{i} + \left(\frac{3}{\sqrt{10}} + \frac{1}{\sqrt{10}}\right)\hat{j} = \frac{4}{\sqrt{10}}\hat{i} + \frac{4}{\sqrt{10}}\hat{j} \]

Thus, the vector \( \mathbf{C} \) can be written as:
\[ \mathbf{C} = k\left(\frac{4}{\sqrt{10}}\hat{i} + \frac{4}{\sqrt{10}}\hat{j}\right) \]

### Step 4: Simplify the Vector
Choosing \( k = \sqrt{10}/4 \) for simplicity, we can express \( \mathbf{C} \) as:
\[ \mathbf{C} = \hat{i} + \hat{j} \]

### Final Answer
The vector equally inclined to \( \hat{i} + 3\hat{j} \) and \( 3\hat{i} + \hat{j} \) is:
\[ \hat{i} + \hat{j} \]