The magnitude of the x-component of vector `vec(A)` is 3 and the magnitude of vector `vec(A)` is 5. What is the magnitude of the y-component of vector `vec(A)` ?

Options

A. 3
B. 4
C. 5
D. 8

Explanation

To find the magnitude of the y-component of vector \(\vec{A}\), we can use the relationship between the components of the vector and its magnitude. Here are the steps to solve the problem:

### Step-by-Step Solution:

1. **Identify the components and magnitude**:
- The x-component of vector \(\vec{A}\) is given as \(A_x = 3\).
- The magnitude of vector \(\vec{A}\) is given as \(|\vec{A}| = 5\).
- We need to find the y-component \(A_y\).

2. **Use the formula for the magnitude of a vector**:
The magnitude of a vector in two dimensions can be expressed as:
\[ |\vec{A}| = \sqrt{A_x^2 + A_y^2} \]

3. **Substitute the known values into the formula**:
Plugging in the values we have:
\[ 5 = \sqrt{3^2 + A_y^2} \]

4. **Square both sides to eliminate the square root**:
\[ 5^2 = 3^2 + A_y^2 \]
This simplifies to:
\[ 25 = 9 + A_y^2 \]

5. **Rearrange the equation to solve for \(A_y^2\)**:
\[ A_y^2 = 25 - 9 \]
\[ A_y^2 = 16 \]

6. **Take the square root to find \(A_y\)**:
\[ A_y = \sqrt{16} = 4 \]

7. **Conclusion**:
The magnitude of the y-component of vector \(\vec{A}\) is \(A_y = 4\).

### Final Answer:
The magnitude of the y-component of vector \(\vec{A}\) is \(4\).

---