If `vec(A)=2hat(i)+hat(j)+hat(k)` and `vec(B)=hat(i)+hat(j)+hat(k)` are two vectors, then the unit vector is

Options

A. Perpendicular to `vec(A)` is `(-hat(j)+hat(k))1/sqrt(2)`
B. Parallel to `vec(A)` is `(2hat(i)+hat(j)+hat(k))/sqrt(6)`
C. Perpendicular to `vec(B)` is `((-hat(j)+hat(k))/sqrt(2))`
D. Parallel to `vec(A)` is `(hat(i)+hat(j)+hat(k))/sqrt(3)`

Explanation

To find the unit vector of the given vector \(\vec{A}\), we can follow these steps:

### Step 1: Identify the vector
Given:
\[ \vec{A} = 2\hat{i} + \hat{j} + \hat{k} \]

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

Calculating the magnitude:
\[ |\vec{A}| = \sqrt{2^2 + 1^2 + 1^2} = \sqrt{4 + 1 + 1} = \sqrt{6} \]

### Step 3: Find the unit vector \(\hat{A}\)
The unit vector \(\hat{A}\) in the direction of \(\vec{A}\) is given by:
\[ \hat{A} = \frac{\vec{A}}{|\vec{A}|} \]
Substituting the values:
\[ \hat{A} = \frac{2\hat{i} + \hat{j} + \hat{k}}{\sqrt{6}} \]

### Step 4: Write the final expression for the unit vector
Thus, the unit vector \(\hat{A}\) is:
\[ \hat{A} = \frac{2\hat{i} + \hat{j} + \hat{k}}{\sqrt{6}} \]

### Final Answer
The unit vector \(\hat{A}\) is:
\[ \hat{A} = \frac{2\hat{i} + \hat{j} + \hat{k}}{\sqrt{6}} \]

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