The unit vector of a vector `vecA =2hati + 4hatj` is

Options

A. `9/sqrt20(hati+hatj)`
B. `sqrt20(2hati+3hatj)`
C. `1/sqrt20(2hati+4hatj)`
D. `9sqrt20(hati+hatj)`

Answer

To find the unit vector of the vector \(\vec{A} = 2\hat{i} + 4\hat{j}\), we will follow these steps:

### Step 1: Find the Magnitude of the Vector \(\vec{A}\)

The magnitude of a vector \(\vec{A} = a\hat{i} + b\hat{j}\) is given by the formula:

\[ |\vec{A}| = \sqrt{a^2 + b^2} \]

For our vector \(\vec{A} = 2\hat{i} + 4\hat{j}\):

\[ |\vec{A}| = \sqrt{(2)^2 + (4)^2} = \sqrt{4 + 16} = \sqrt{20} \]

### Step 2: Calculate the Unit Vector

The unit vector \(\hat{A}\) in the direction of \(\vec{A}\) is given by:

\[ \hat{A} = \frac{\vec{A}}{|\vec{A}|} \]

Substituting the values we have:

\[ \hat{A} = \frac{2\hat{i} + 4\hat{j}}{\sqrt{20}} \]

### Step 3: Simplify the Unit Vector

We can simplify the expression for the unit vector:

\[ \hat{A} = \frac{2}{\sqrt{20}}\hat{i} + \frac{4}{\sqrt{20}}\hat{j} \]

### Step 4: Further Simplification

We can simplify \(\sqrt{20}\):

\[ \sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5} \]

Thus, we can rewrite the unit vector:

\[ \hat{A} = \frac{2}{2\sqrt{5}}\hat{i} + \frac{4}{2\sqrt{5}}\hat{j} = \frac{1}{\sqrt{5}}\hat{i} + \frac{2}{\sqrt{5}}\hat{j} \]

### Final Answer

The unit vector \(\hat{A}\) is:

\[ \hat{A} = \frac{1}{\sqrt{5}}\hat{i} + \frac{2}{\sqrt{5}}\hat{j} \]