Given vector`overset(rarr)A=2 hat I +3hatj` , the angle between `overset(rarr)A` and y-axis is :

Options

A. `tan^(-1)3//2`
B. `tan^(-1) 2//32`
C. `sin^(-1)2//3`
D. `cos^(-1)2//3`

Explanation

To find the angle between the vector \(\overset{\rarr}{A} = 2\hat{i} + 3\hat{j}\) and the y-axis, we can follow these steps:

### Step 1: Identify the components of the vector
The vector \(\overset{\rarr}{A}\) has two components:
- The x-component \(A_x = 2\) (coefficient of \(\hat{i}\))
- The y-component \(A_y = 3\) (coefficient of \(\hat{j}\))

### Step 2: Understand the relationship between the components and the angle
The angle \(\theta\) between the vector and the y-axis can be found using the tangent function. The tangent of the angle is given by the ratio of the opposite side (the x-component) to the adjacent side (the y-component).

### Step 3: Write the formula for tangent
The formula for the tangent of the angle \(\theta\) is:
\[ \tan(\theta) = \frac{A_x}{A_y} \]
Substituting the values we have:
\[ \tan(\theta) = \frac{2}{3} \]

### Step 4: Calculate the angle using the inverse tangent function
To find the angle \(\theta\), we take the inverse tangent (arctan) of the ratio:
\[ \theta = \tan^{-1}\left(\frac{2}{3}\right) \]

### Step 5: Conclusion
Thus, the angle between the vector \(\overset{\rarr}{A}\) and the y-axis is:
\[ \theta = \tan^{-1}\left(\frac{2}{3}\right) \]