The image you've uploaded contains a mathematical problem involving vectors, specifically asking for the magnitude and direction of a resultant vector and the angle between two vectors. Let's break down the problem step by step.

Problem Analysis

Given:

• Vector A with a magnitude of $|\mathbf{A}| = 4 \, \text{N}$ at an angle of $30^\circ$ above the x-axis.
• Vector B with a magnitude of $|\mathbf{B}| = 3 \, \text{N}$ at an angle of $60^\circ$ above the x-axis.

Tasks:

1. Determine the magnitude and direction of the resultant vector $\mathbf{R} = \mathbf{A} + \mathbf{B}$.
2. Determine the angle between $\mathbf{A}$ and $\mathbf{B}$.

Step 1: Calculate the Resultant Vector $\mathbf{R}$

To find the resultant vector $\mathbf{R}$, we'll break both vectors $\mathbf{A}$ and $\mathbf{B}$ into their respective components.

For vector $\mathbf{A}$:

• $A_x = |\mathbf{A}| \cos(30^\circ) = 4 \times \cos(30^\circ)$
• $A_y = |\mathbf{A}| \sin(30^\circ) = 4 \times \sin(30^\circ)$

For vector $\mathbf{B}$:

• $B_x = |\mathbf{B}| \cos(60^\circ) = 3 \times \cos(60^\circ)$
• $B_y = |\mathbf{B}| \sin(60^\circ) = 3 \times \sin(60^\circ)$

The components of the resultant vector $\mathbf{R}$ are then:

• $R_x = A_x + B_x$
• $R_y = A_y + B_y$

The magnitude of $\mathbf{R}$ is: $|\mathbf{R}| = \sqrt{R_x^2 + R_y^2}$

And the direction $\theta_R$ of the resultant vector $\mathbf{R}$ is: $\theta_R = \tan^{-1}\left(\frac{R_y}{R_x}\right)$

Step 2: Calculate the Angle Between $\mathbf{A}$ and $\mathbf{B}$

The angle $\theta$ between the two vectors can be found using the dot product formula: $\mathbf{A} \cdot \mathbf{B} = |\mathbf{A}| |\mathbf{B}| \cos(\theta)$ Where: $\cos(\theta) = \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{A}| |\mathbf{B}|}$

The dot product $\mathbf{A} \cdot \mathbf{B}$ can also be calculated as: $\mathbf{A} \cdot \mathbf{B} = A_x B_x + A_y B_y$

Finally, solve for $\theta$: $\theta = \cos^{-1}\left(\frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{A}| |\mathbf{B}|}\right)$

Would you like to proceed with the calculations, or do you need clarification on any part of the process?

Follow-up Questions

1. What are the components of vector $\mathbf{A}$ and vector $\mathbf{B}$ along the x-axis and y-axis?
2. How does the angle between two vectors affect their resultant vector?
3. How can you determine if two vectors are perpendicular using the dot product?
4. What would be the magnitude of the resultant vector if the two vectors were aligned along the same direction?
5. What changes in calculations if the vectors $\mathbf{A}$ and $\mathbf{B}$ were given in unit vector form?

Tip

Always ensure that angles are correctly resolved into their components along the axes when working with vectors, as mistakes in this step can lead to incorrect results in further calculations.