What is the vector whose magnitude is 3, and is perpendicular to `hat(i)+hat(j) and hat(j)+hat(k)`?

Options

A. `3(vec(i)+hat(j)-vec(k))`
B. `sqrt(3)(vec(i)-vec(j)+vec(k))`
C. `sqrt(3)(vec(i)+vec(j)+vec(k))`
D. `3(vec(i)-vec(j)+vec(k))`

Explanation

To find the vector whose magnitude is 3 and is perpendicular to the vectors \( \hat{i} + \hat{j} \) and \( \hat{j} + \hat{k} \), we can follow these steps:

### Step 1: Define the Required Vector
Let the required vector be represented as:
\[ \mathbf{r} = x \hat{i} + y \hat{j} + z \hat{k} \]

### Step 2: Set Up the Perpendicularity Conditions
Since the vector \( \mathbf{r} \) is perpendicular to both \( \hat{i} + \hat{j} \) and \( \hat{j} + \hat{k} \), we can use the dot product to set up our equations:
1. \( \mathbf{r} \cdot (\hat{i} + \hat{j}) = 0 \)
2. \( \mathbf{r} \cdot (\hat{j} + \hat{k}) = 0 \)

### Step 3: Calculate the Dot Products
Calculating the first dot product:
\[ \mathbf{r} \cdot (\hat{i} + \hat{j}) = (x \hat{i} + y \hat{j} + z \hat{k}) \cdot (\hat{i} + \hat{j}) = x + y = 0 \]
This gives us our first equation:
\[ x + y = 0 \quad \text{(1)} \]

Now calculating the second dot product:
\[ \mathbf{r} \cdot (\hat{j} + \hat{k}) = (x \hat{i} + y \hat{j} + z \hat{k}) \cdot (\hat{j} + \hat{k}) = y + z = 0 \]
This gives us our second equation:
\[ y + z = 0 \quad \text{(2)} \]

### Step 4: Express Variables in Terms of One Another
From equation (1), we can express \( x \) in terms of \( y \):
\[ x = -y \]
From equation (2), we can express \( z \) in terms of \( y \):
\[ z = -y \]

### Step 5: Use the Magnitude Condition
We know the magnitude of \( \mathbf{r} \) is 3:
\[ |\mathbf{r}| = \sqrt{x^2 + y^2 + z^2} = 3 \]
Squaring both sides gives:
\[ x^2 + y^2 + z^2 = 9 \]

### Step 6: Substitute for \( x \) and \( z \)
Substituting \( x = -y \) and \( z = -y \) into the magnitude equation:
\[ (-y)^2 + y^2 + (-y)^2 = 9 \]
This simplifies to:
\[ y^2 + y^2 + y^2 = 9 \implies 3y^2 = 9 \]
Thus,
\[ y^2 = 3 \implies y = \pm \sqrt{3} \]

### Step 7: Find \( x \) and \( z \)
Now substituting back to find \( x \) and \( z \):
- If \( y = \sqrt{3} \):
- \( x = -\sqrt{3} \)
- \( z = -\sqrt{3} \)

- If \( y = -\sqrt{3} \):
- \( x = \sqrt{3} \)
- \( z = \sqrt{3} \)

### Step 8: Write the Vectors
Thus, we have two possible vectors:
1. \( \mathbf{r_1} = -\sqrt{3} \hat{i} + \sqrt{3} \hat{j} - \sqrt{3} \hat{k} \)
2. \( \mathbf{r_2} = \sqrt{3} \hat{i} - \sqrt{3} \hat{j} + \sqrt{3} \hat{k} \)

### Conclusion
The required vectors are:
\[ \mathbf{r_1} = -\sqrt{3} \hat{i} + \sqrt{3} \hat{j} - \sqrt{3} \hat{k} \quad \text{or} \quad \mathbf{r_2} = \sqrt{3} \hat{i} - \sqrt{3} \hat{j} + \sqrt{3} \hat{k} \]