To solve the problem step by step, we need to find a vector \( \vec{r} \) of magnitude 4 that is equally inclined to the vectors \( \hat{i} + \hat{j} \), \( \hat{j} + \hat{k} \), and \( \hat{k} + \hat{i} \).
### Step 1: Define the Required Vector
Let the required vector be represented as:
\[
\vec{r} = x \hat{i} + y \hat{j} + z \hat{k}
\]
### Step 2: Set the Magnitude of the Vector
The magnitude of the vector \( \vec{r} \) is given as 4, so we have:
\[
|\vec{r}| = \sqrt{x^2 + y^2 + z^2} = 4
\]
Squaring both sides, we get:
\[
x^2 + y^2 + z^2 = 16
\]
### Step 3: Equally Inclined Condition
Since \( \vec{r} \) is equally inclined to the vectors \( \hat{i} + \hat{j} \), \( \hat{j} + \hat{k} \), and \( \hat{k} + \hat{i} \), we can express this condition mathematically. The angle between two vectors \( \vec{a} \) and \( \vec{b} \) can be found using the dot product:
\[
\cos \theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}| |\vec{b}|}
\]
For our case, we can set up the following equations based on the condition of equal inclination.
1. For \( \hat{i} + \hat{j} \):
\[
\frac{\vec{r} \cdot (\hat{i} + \hat{j})}{|\vec{r}| \cdot |\hat{i} + \hat{j}|} = \cos \theta
\]
2. For \( \hat{j} + \hat{k} \):
\[
\frac{\vec{r} \cdot (\hat{j} + \hat{k})}{|\vec{r}| \cdot |\hat{j} + \hat{k}|} = \cos \theta
\]
3. For \( \hat{k} + \hat{i} \):
\[
\frac{\vec{r} \cdot (\hat{k} + \hat{i})}{|\vec{r}| \cdot |\hat{k} + \hat{i}|} = \cos \theta
\]
### Step 4: Set Up the Equations
Calculating the dot products, we have:
1. \( \vec{r} \cdot (\hat{i} + \hat{j}) = x + y \)
2. \( \vec{r} \cdot (\hat{j} + \hat{k}) = y + z \)
3. \( \vec{r} \cdot (\hat{k} + \hat{i}) = z + x \)
Since all three expressions are equal to some constant \( \lambda \), we can write:
\[
x + y = y + z = z + x = \lambda
\]
### Step 5: Solve for Variables
From the equations \( x + y = \lambda \), \( y + z = \lambda \), and \( z + x = \lambda \), we can derive:
- From \( x + y = \lambda \) and \( y + z = \lambda \), we can express:
\[
x + y = y + z \implies x = z
\]
- Similarly, from \( y + z = \lambda \) and \( z + x = \lambda \):
\[
y + z = z + x \implies y = x
\]
Thus, we can conclude:
\[
x = y = z
\]
### Step 6: Substitute Back
Let \( x = y = z = k \). Then:
\[
3k^2 = 16 \implies k^2 = \frac{16}{3} \implies k = \frac{4}{\sqrt{3}} \text{ or } -\frac{4}{\sqrt{3}}
\]
### Step 7: Final Vector
Thus, the vector \( \vec{r} \) can be expressed as:
\[
\vec{r} = k(\hat{i} + \hat{j} + \hat{k}) = \frac{4}{\sqrt{3}}(\hat{i} + \hat{j} + \hat{k})
\]
### Conclusion
The required vector of magnitude 4 which is equally inclined to the vectors \( \hat{i} + \hat{j} \), \( \hat{j} + \hat{k} \), and \( \hat{k} + \hat{i} \) is:
\[
\vec{r} = \frac{4}{\sqrt{3}}(\hat{i} + \hat{j} + \hat{k})
\]
