Write a vector of magnitude of 18 units in the direction of the vector `hati-2hatj-2hatk`.

Answer

To find a vector of magnitude 18 units in the direction of the vector \( \hat{i} - 2\hat{j} - 2\hat{k} \), we can follow these steps:

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

### Step 2: Calculate the magnitude of the given vector
The magnitude \( |\mathbf{v}| \) of the vector \( \mathbf{v} = a\hat{i} + b\hat{j} + c\hat{k} \) is given by the formula:
\[ |\mathbf{v}| = \sqrt{a^2 + b^2 + c^2} \]
For our vector:
- \( a = 1 \)
- \( b = -2 \)
- \( c = -2 \)

Calculating the magnitude:
\[ |\mathbf{v}| = \sqrt{1^2 + (-2)^2 + (-2)^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3 \]

### Step 3: Find the unit vector in the direction of \( \mathbf{v} \)
The unit vector \( \hat{u} \) in the direction of \( \mathbf{v} \) is given by:
\[ \hat{u} = \frac{\mathbf{v}}{|\mathbf{v}|} \]
Substituting the values:
\[ \hat{u} = \frac{\hat{i} - 2\hat{j} - 2\hat{k}}{3} = \frac{1}{3}\hat{i} - \frac{2}{3}\hat{j} - \frac{2}{3}\hat{k} \]

### Step 4: Scale the unit vector to the desired magnitude
To find a vector of magnitude 18 in the direction of \( \hat{u} \), we multiply the unit vector by 18:
\[ \mathbf{v}_{18} = 18 \cdot \hat{u} = 18 \left( \frac{1}{3}\hat{i} - \frac{2}{3}\hat{j} - \frac{2}{3}\hat{k} \right) \]
Calculating this gives:
\[ \mathbf{v}_{18} = 6\hat{i} - 12\hat{j} - 12\hat{k} \]

### Final Answer
The vector of magnitude 18 units in the direction of \( \hat{i} - 2\hat{j} - 2\hat{k} \) is:
\[ \mathbf{v}_{18} = 6\hat{i} - 12\hat{j} - 12\hat{k} \]

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