The units vectors orthogonal to the vector `- hat i + 2hat j + 2hat k` and making equal angles with the X and Y axes islare) :

Options

A. `+-1/3(2hati+2hatj-hatk)`
B. `19/(5sqrt43)`
C. `+-1/3(hati+hatj-hatk)`
D. none of these

Explanation

To find the unit vectors orthogonal to the vector \(-\hat{i} + 2\hat{j} + 2\hat{k}\) and making equal angles with the X and Y axes, we can follow these steps:

### Step 1: Define the Unit Vector
Let the required unit vector be represented as:
\[ \vec{R} = l\hat{i} + m\hat{j} + n\hat{k} \]
where \(l\), \(m\), and \(n\) are the direction cosines of the vector.

### Step 2: Use the Unit Vector Condition
Since \(\vec{R}\) is a unit vector, it must satisfy the equation:
\[ l^2 + m^2 + n^2 = 1 \tag{1} \]

### Step 3: Equal Angles with X and Y Axes
The problem states that the vector makes equal angles with the X and Y axes, which implies:
\[ l = m \tag{2} \]

### Step 4: Substitute Equation (2) into Equation (1)
Substituting \(m = l\) into equation (1):
\[ l^2 + l^2 + n^2 = 1 \]
This simplifies to:
\[ 2l^2 + n^2 = 1 \tag{3} \]

### Step 5: Orthogonality Condition
The vector \(\vec{R}\) is orthogonal to the vector \(-\hat{i} + 2\hat{j} + 2\hat{k}\). Therefore, their dot product must be zero:
\[ \vec{R} \cdot (-\hat{i} + 2\hat{j} + 2\hat{k}) = 0 \]
Calculating the dot product:
\[ l(-1) + m(2) + n(2) = 0 \]
Substituting \(m = l\):
\[ -l + 2l + 2n = 0 \]
This simplifies to:
\[ l + 2n = 0 \tag{4} \]

### Step 6: Substitute Equation (4) into Equation (3)
From equation (4), we can express \(n\) in terms of \(l\):
\[ n = -\frac{l}{2} \]
Now substitute this into equation (3):
\[ 2l^2 + \left(-\frac{l}{2}\right)^2 = 1 \]
This simplifies to:
\[ 2l^2 + \frac{l^2}{4} = 1 \]
Multiplying through by 4 to eliminate the fraction:
\[ 8l^2 + l^2 = 4 \]
Thus:
\[ 9l^2 = 4 \implies l^2 = \frac{4}{9} \implies l = \pm \frac{2}{3} \]

### Step 7: Find \(m\) and \(n\)
Since \(m = l\):
\[ m = \pm \frac{2}{3} \]
Now substituting \(l\) back into equation (4) to find \(n\):
\[ n = -\frac{l}{2} = -\frac{\pm \frac{2}{3}}{2} = \mp \frac{1}{3} \]

### Step 8: Write the Final Unit Vectors
Thus, the unit vectors orthogonal to the given vector and making equal angles with the X and Y axes are:
\[ \vec{R} = \pm \left(\frac{2}{3}\hat{i} + \frac{2}{3}\hat{j} - \frac{1}{3}\hat{k}\right) \]

### Final Answer
The required unit vectors are:
\[ \frac{2}{3}\hat{i} + \frac{2}{3}\hat{j} - \frac{1}{3}\hat{k} \quad \text{and} \quad -\left(\frac{2}{3}\hat{i} + \frac{2}{3}\hat{j} - \frac{1}{3}\hat{k}\right) \]