We are given two vectors $\mathbf{A}$ and $\mathbf{B}$, each with magnitude 4.92, and their sum results in $7.14\hat{j}$. We need to determine the angle between them.

Step 1: Represent the vectors in components

Assume the two vectors have equal magnitudes and form an angle $\theta$ with each other.

Let $\mathbf{A}$ and $\mathbf{B}$ have the following forms: $\mathbf{A} = 4.92(\cos\theta \hat{i} + \sin\theta \hat{j})$ $\mathbf{B} = 4.92(\cos\phi \hat{i} + \sin\phi \hat{j})$ Since $\mathbf{A} + \mathbf{B} = 7.14\hat{j}$, we know the $i$-components cancel each other out and the $j$-components sum to 7.14.

Step 2: Set up the equations

• For the $i$-components: $4.92\cos\theta + 4.92\cos\phi = 0 \quad \Rightarrow \quad \cos\theta + \cos\phi = 0$ Since $\theta$ and $\phi$ are symmetric, $\theta = -\phi$, so $\cos\theta = -\cos\phi$.

• For the $j$-components: $4.92\sin\theta + 4.92\sin\phi = 7.14 \quad \Rightarrow \quad \sin\theta + \sin\phi = \frac{7.14}{4.92} = 1.45$ Since $\theta = -\phi$, we have $\sin\theta = \sin\phi$. Thus, we can write: $2\sin\theta = 1.45 \quad \Rightarrow \quad \sin\theta = 0.725$

Step 3: Solve for $\theta$

Taking the inverse sine of both sides: $\theta = \sin^{-1}(0.725) \approx 46.6^\circ$

Thus, the angle between the two vectors is $2\theta = 2 \times 46.6^\circ = 93.2^\circ$.

Final Answer:

The angle between vectors $\mathbf{A}$ and $\mathbf{B}$ is approximately $93.2^\circ$.

Would you like more details on any step? Here are some follow-up questions to further explore:

1. How does vector addition work in component form?
2. What are other ways to find angles between vectors?
3. How can we apply the dot product to solve similar problems?
4. How does the inverse sine function behave in different quadrants?
5. What is the geometric interpretation of adding two vectors?

Tip: When working with vector addition, always break the vectors into their components to simplify the process of solving.