Given two vectors $\mathbf{A}$ and $\mathbf{B}$, both having equal magnitudes of 4.75, and their sum is $\mathbf{A} + \mathbf{B} = 7.59 \hat{j}$, we are tasked with finding the angle $\theta$ between these two vectors.

Step 1: Express the vectors

The vectors $\mathbf{A}$ and $\mathbf{B}$ can be written in terms of their components:

$\mathbf{A} = 4.75 (\cos \theta_1 \hat{i} + \sin \theta_1 \hat{j})$ $\mathbf{B} = 4.75 (\cos \theta_2 \hat{i} + \sin \theta_2 \hat{j})$

Since both vectors have the same magnitude, let's assume without loss of generality that one vector is at an angle $\theta/2$ above the horizontal and the other is at an angle $-\theta/2$ below the horizontal:

$\mathbf{A} = 4.75 (\cos(\theta/2) \hat{i} + \sin(\theta/2) \hat{j})$ $\mathbf{B} = 4.75 (\cos(\theta/2) \hat{i} - \sin(\theta/2) \hat{j})$

Step 2: Sum of the vectors

The sum $\mathbf{A} + \mathbf{B}$ will have its $\hat{i}$-components cancel out because of symmetry, leaving only the $\hat{j}$-components:

$\mathbf{A} + \mathbf{B} = 4.75 \sin(\theta/2) \hat{j} + 4.75 \sin(\theta/2) \hat{j} = 9.5 \sin(\theta/2) \hat{j}$

Step 3: Set the sum equal to the given value

We are given that the sum of the vectors is $7.59 \hat{j}$, so we can set the magnitudes of the $\hat{j}$-components equal:

$9.5 \sin(\theta/2) = 7.59$

Step 4: Solve for $\theta/2$

Solving for $\sin(\theta/2)$:

$\sin(\theta/2) = \frac{7.59}{9.5} = 0.7989$

Now, take the inverse sine to find $\theta/2$:

$\theta/2 = \sin^{-1}(0.7989) \approx 53.01^\circ$

Thus,

$\theta \approx 2 \times 53.01^\circ = 106.02^\circ$

Final Answer:

The angle between vectors $\mathbf{A}$ and $\mathbf{B}$ is approximately $\boxed{106.02^\circ}$.

Would you like further details or clarifications?

Related Questions:

1. How do you find the dot product of two vectors when their angle is known?
2. What is the geometric interpretation of two vectors with the same magnitude?
3. How do you break down vectors into components for different coordinate systems?
4. What happens when two vectors have an angle of $90^\circ$ between them?
5. How does the vector sum change if the angle between the vectors is zero?

Tip:

When adding vectors geometrically, always break them down into components, as it simplifies the process of solving vector equations.