value of a unit vector in the direction of vector `A=5 hat i- 12 hat j` is :

Options

A. `hati`
B. `hat j`
C. `(hat i+hat j)/(13)`
D. `(5hat i-12hat j)/(13)`

Explanation

To find the unit vector in the direction of the vector \( A = 5 \hat{i} - 12 \hat{j} \), we can follow these steps:

### Step 1: Write down the vector
The given vector is:
\[ A = 5 \hat{i} - 12 \hat{j} \]

### Step 2: Calculate the magnitude of the vector
The magnitude of vector \( A \) is calculated using the formula:
\[ |A| = \sqrt{(A_x)^2 + (A_y)^2} \]
where \( A_x \) and \( A_y \) are the components of the vector along the \( \hat{i} \) and \( \hat{j} \) directions, respectively.

For our vector:
\[ |A| = \sqrt{(5)^2 + (-12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \]

### Step 3: Calculate the unit vector
The unit vector \( \hat{A} \) in the direction of vector \( A \) is given by:
\[ \hat{A} = \frac{A}{|A|} \]
Substituting the values we have:
\[ \hat{A} = \frac{5 \hat{i} - 12 \hat{j}}{13} \]

### Step 4: Write the final answer
Thus, the unit vector in the direction of vector \( A \) is:
\[ \hat{A} = \frac{5}{13} \hat{i} - \frac{12}{13} \hat{j} \]

### Summary
The value of the unit vector in the direction of vector \( A = 5 \hat{i} - 12 \hat{j} \) is:
\[ \hat{A} = \frac{5}{13} \hat{i} - \frac{12}{13} \hat{j} \]

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