Find a unit vector in the direction of vector `vecb = hati + 2hatj + 3hatk`.

Options

A. `(1)/(sqrt(14))hati+(2)/(sqrt(14))hatj+(3)/(sqrt(14))hatk`
B. `(2)/(sqrt(14))hati+(1)/(sqrt(14))hatj+(3)/(sqrt(14))hatk`
C. `(1)/(sqrt(14))hati+(3)/(sqrt(14))hatj+(2)/(sqrt(14))hatk`
D. `(3)/(sqrt(14))hati+(1)/(sqrt(14))hatj+(2)/(sqrt(14))hatk`

Answer

To find a unit vector in the direction of the vector \(\vec{b} = \hat{i} + 2\hat{j} + 3\hat{k}\), we will follow these steps:

### Step 1: Identify the vector
The vector given is:
\[ \vec{b} = \hat{i} + 2\hat{j} + 3\hat{k} \]

### Step 2: Calculate the magnitude of the vector
The magnitude of a vector \(\vec{b} = a\hat{i} + b\hat{j} + c\hat{k}\) is calculated using the formula:
\[ |\vec{b}| = \sqrt{a^2 + b^2 + c^2} \]
For our vector:
- \(a = 1\) (coefficient of \(\hat{i}\))
- \(b = 2\) (coefficient of \(\hat{j}\))
- \(c = 3\) (coefficient of \(\hat{k}\))

Thus, the magnitude is:
\[ |\vec{b}| = \sqrt{1^2 + 2^2 + 3^2} = \sqrt{1 + 4 + 9} = \sqrt{14} \]

### Step 3: Calculate the unit vector
The unit vector \(\hat{b}\) in the direction of \(\vec{b}\) is given by:
\[ \hat{b} = \frac{\vec{b}}{|\vec{b}|} \]
Substituting the values we have:
\[ \hat{b} = \frac{\hat{i} + 2\hat{j} + 3\hat{k}}{\sqrt{14}} \]

### Step 4: Write the unit vector in component form
We can express the unit vector as:
\[ \hat{b} = \frac{1}{\sqrt{14}}\hat{i} + \frac{2}{\sqrt{14}}\hat{j} + \frac{3}{\sqrt{14}}\hat{k} \]

### Final Answer
Thus, the unit vector in the direction of \(\vec{b}\) is:
\[ \hat{b} = \frac{1}{\sqrt{14}}\hat{i} + \frac{2}{\sqrt{14}}\hat{j} + \frac{3}{\sqrt{14}}\hat{k} \]

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