In this problem, you are tasked with finding the components of vector D based on the equation:

$\vec{D} = 4.10 (\vec{A} + \vec{B})$

where vectors $\vec{A}$ and $\vec{B}$ are given in the figure:

• $\vec{A}$ has a magnitude of 2.0 m and is oriented 15° above the positive y-axis.
• $\vec{B}$ has a magnitude of 4.0 m and is oriented 15° below the positive x-axis.

Step 1: Resolve $\vec{A}$ and $\vec{B}$ into their components

For $\vec{A}$:

• The x-component $A_x$ and y-component $A_y$ can be found using trigonometry: $A_x = A \sin(15^\circ)$ $A_y = A \cos(15^\circ)$ Given $A = 2.0 \, \text{m}$: $A_x = 2.0 \sin(15^\circ), \quad A_y = 2.0 \cos(15^\circ)$

For $\vec{B}$:

• Similarly, the x-component $B_x$ and y-component $B_y$: $B_x = B \cos(15^\circ)$ $B_y = -B \sin(15^\circ)$ Given $B = 4.0 \, \text{m}$, and noting the negative sign for $B_y$ because it is below the x-axis: $B_x = 4.0 \cos(15^\circ), \quad B_y = -4.0 \sin(15^\circ)$

Step 2: Add the components

Now, sum the components of $\vec{A}$ and $\vec{B}$: $A_x + B_x, \quad A_y + B_y$

Step 3: Multiply by 4.10

Finally, multiply both the x and y components by 4.10: $D_x = 4.10 \times (A_x + B_x), \quad D_y = 4.10 \times (A_y + B_y)$

Let me calculate the exact values for you.The components of vector $\vec{D}$ are:

$D_x \approx 17.96 \, \text{m}, \quad D_y \approx 3.68 \, \text{m}$

Would you like more details or have any further questions?

Here are some related questions to explore:

1. How do you calculate vector components given an angle and magnitude?
2. What is the significance of negative vector components in physics problems?
3. How would you add more than two vectors using their components?
4. What are some real-world applications of vector addition?
5. How does changing the angle of a vector affect its components?

Tip: When dealing with vectors, always make sure to resolve them correctly into components before performing any operations like addition or multiplication!