The vector in the direction of the vector `hati-2hatj+2hatk` that has magnitude 9 is

Options

A. `hati-2hatj+2hatk`
B. `(hati-2hatj+2hatk)/(3)`
C. `3(hati-2hatj+2hatk)`
D. `9(hati-2hatj+2hatk)`

Explanation

To find the vector in the direction of the vector \( \hat{i} - 2\hat{j} + 2\hat{k} \) that has a magnitude of 9, we can follow these steps:

### Step 1: Identify the given vector
The given vector is:
\[ \mathbf{A} = \hat{i} - 2\hat{j} + 2\hat{k} \]

### Step 2: Calculate the magnitude of the vector
The magnitude of vector \( \mathbf{A} \) is calculated using the formula:
\[ |\mathbf{A}| = \sqrt{(1)^2 + (-2)^2 + (2)^2} \]
Calculating this gives:
\[ |\mathbf{A}| = \sqrt{1 + 4 + 4} = \sqrt{9} = 3 \]

### Step 3: Find the unit vector in the direction of \( \mathbf{A} \)
The unit vector \( \hat{u} \) in the direction of \( \mathbf{A} \) is given by:
\[ \hat{u} = \frac{\mathbf{A}}{|\mathbf{A}|} = \frac{\hat{i} - 2\hat{j} + 2\hat{k}}{3} \]

### Step 4: Scale the unit vector to the desired magnitude
To find the vector in the direction of \( \mathbf{A} \) with a magnitude of 9, we multiply the unit vector by 9:
\[ \mathbf{B} = 9 \cdot \hat{u} = 9 \cdot \frac{\hat{i} - 2\hat{j} + 2\hat{k}}{3} \]
This simplifies to:
\[ \mathbf{B} = 3(\hat{i} - 2\hat{j} + 2\hat{k}) = 3\hat{i} - 6\hat{j} + 6\hat{k} \]

### Final Answer
The vector in the direction of \( \hat{i} - 2\hat{j} + 2\hat{k} \) that has a magnitude of 9 is:
\[ \mathbf{B} = 3\hat{i} - 6\hat{j} + 6\hat{k} \]