The angle which the vector `vec(A)=2hat(i)+3hat(j)` makes with the y-axis, where `hat(i)` and `hat(j)` are unit vectors along x- and y-axis, respectively, is

Options

A. `cos^(-1)(3//5)`
B. `cos^(-1)(2//3)`
C. `tan^(-1)(2//3)
D. `sin^(-1)(2//3)`

Explanation

To find the angle that the vector \(\vec{A} = 2\hat{i} + 3\hat{j}\) makes with the y-axis, we can follow these steps:

### Step 1: Understand the components of the vector
The vector \(\vec{A}\) has two components:
- The x-component is \(A_x = 2\)
- The y-component is \(A_y = 3\)

### Step 2: Use the definition of the angle with respect to the y-axis
The angle \(\theta\) that the vector makes with the y-axis can be found using the tangent function:
\[ \tan(\theta) = \frac{A_x}{A_y} \]
This is because the tangent of the angle is the ratio of the opposite side (x-component) to the adjacent side (y-component) in the right triangle formed by the vector.

### Step 3: Substitute the values into the equation
Substituting the values of \(A_x\) and \(A_y\):
\[ \tan(\theta) = \frac{2}{3} \]

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

### Step 5: Use a calculator to find the angle
Using a calculator, we can find:
\[ \theta \approx 33.69^\circ \]

### Final Answer
Thus, the angle which the vector \(\vec{A} = 2\hat{i} + 3\hat{j}\) makes with the y-axis is approximately \(33.69^\circ\).

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