The vector in the direction of the vector `hat(i) - 2 hat(j) + 2hat(k)` that has magnitude 9 units is

Options

A. `hat(i) - 2 hat(j) + 2 hat(k)`
B. `(hat(i) - 2 hat(j) + 2 hat(k))/(3)`
C. `3( hat(i)- 2 hat(j) + 2 hat(k))`
D. `9 ( hat(i) - 2 hat(j) + 2 hat(k))`

Answer

To find the vector in the direction of the vector \(\hat{i} - 2\hat{j} + 2\hat{k}\) that has a magnitude of 9 units, 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 \(|\mathbf{a}|\) of the vector \(\mathbf{a}\) can be calculated using the formula:
\[ |\mathbf{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2} \]
where \(a_1\), \(a_2\), and \(a_3\) are the coefficients of \(\hat{i}\), \(\hat{j}\), and \(\hat{k}\) respectively.

For our vector:
- \(a_1 = 1\)
- \(a_2 = -2\)
- \(a_3 = 2\)

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

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

### Step 4: Scale the unit vector to the desired magnitude
To find the vector with a magnitude of 9 units in the same direction, we multiply the unit vector by 9:
\[ \mathbf{v} = 9 \hat{u} = 9 \left(\frac{1}{3}\hat{i} - \frac{2}{3}\hat{j} + \frac{2}{3}\hat{k}\right) \]
Calculating this gives:
\[ \mathbf{v} = 3\hat{i} - 6\hat{j} + 6\hat{k} \]

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

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