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The problem requires finding vector \(\vec{D} = 4.10 \vec{A} + \vec{B}\), where vectors \(\vec{A}\) and \(\vec{B}\) are given in the image. Let's break this down:

### Step 1: Decompose the vectors \(\vec{A}\) and \(\vec{B}\) into their components.

- Vector \(\vec{A}\) is given as 2.0 m in magnitude and directed at 15° above the positive \(y\)-axis.
  - The components of \(\vec{A}\):
    \[
    A_x = 2.0 \sin(15^\circ) \quad \text{(since it's tilted towards the \(x\)-axis)}
    \]
    \[
    A_y = 2.0 \cos(15^\circ)
    \]

- Vector \(\vec{B}\) is given as 4.0 m in magnitude and directed at 15° below the positive \(x\)-axis.
  - The components of \(\vec{B}\):
    \[
    B_x = 4.0 \cos(15^\circ)
    \]
    \[
    B_y = -4.0 \sin(15^\circ) \quad \text{(since it's below the \(x\)-axis)}
    \]

### Step 2: Calculate the components of \(4.10 \vec{A}\).

Since we are asked for \(4.10 \vec{A}\), we multiply the components of \(\vec{A}\) by 4.10:
\[
(4.10 A_x, 4.10 A_y)
\]

### Step 3: Add the vectors \(\vec{D} = 4.10 \vec{A} + \vec{B}\).

To find the components of \(\vec{D}\), sum the \(x\)- and \(y\)-components of \(4.10 \vec{A}\) and \(\vec{B}\) separately:
\[
D_x = 4.10 A_x + B_x
\]
\[
D_y = 4.10 A_y + B_y
\]

Now, I will calculate the components using these expressions.The components of vector \(\vec{D} = 4.10 \vec{A} + \vec{B}\) are approximately:

\[
D_x = 5.99 \, \text{m}, \quad D_y = 6.89 \, \text{m}
\]

Would you like further details or explanations on any step? Here are some related questions that might expand on this:

1. How are vector components calculated from magnitudes and angles?
2. Why are trigonometric functions used to find vector components?
3. How does the direction of a vector affect the signs of its components?
4. What is the significance of multiplying a vector by a scalar?
5. How can we find the magnitude and direction of the resultant vector \(\vec{D}\)?

**Tip:** When adding vectors, always ensure you resolve them into components to make the process easier and more accurate.

Find the components of vector D = 4.10A + B, where A and B are vectors with known magnitudes and angles.

Learn how to decompose vectors into components and find the resultant vector D = 4.10A + B. This problem involves using trigonometric functions to resolve magnitudes at known angles. Suitable for high school students in Grades 10-12.

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