https://file.aiquickdraw.com/mathbot/file/1726793634440ym7kqw94.jpg

Let's break down and solve the problem step by step.

### Part (a): 
The task is to find the diagram that shows the correct addition of vectors \( \mathbf{A} \) and \( \mathbf{D} \) using the tip-to-tail method. 

Vector \( \mathbf{D} \) is given as \( \mathbf{D} = (2 \, \text{m})\mathbf{i} + (-1 \, \text{m})\mathbf{j} \), meaning it has components:
- \( D_x = 2 \, \text{m} \)
- \( D_y = -1 \, \text{m} \)

Looking at the diagrams:
1. Identify where vector \( D \) is placed relative to vector \( A \). The correct method involves placing the tail of \( D \) at the tip of \( A \). 
2. The resultant vector \( R \) should extend from the origin of \( A \) to the tip of \( D \).

We can identify the correct diagram by finding the one that correctly places vector \( D \) at the tip of \( A \).

### Part (b):
You need to fill in the table with the x and y components of vectors \( A \) and \( D \), and then find the components of their resultant vector \( \mathbf{R} = \mathbf{A} + \mathbf{D} \).

Let's first read the components of vector \( A \) from the graph:
- \( A_x = -4 \, \text{m} \) (since \( A \) extends from 0 to -4 on the x-axis)
- \( A_y = 3 \, \text{m} \) (since \( A \) extends from 0 to 3 on the y-axis)

Vector \( D \) was given as:
- \( D_x = 2 \, \text{m} \)
- \( D_y = -1 \, \text{m} \)

Now calculate the components of the resultant vector \( \mathbf{R} \):
- \( R_x = A_x + D_x = -4 + 2 = -2 \, \text{m} \)
- \( R_y = A_y + D_y = 3 + (-1) = 2 \, \text{m} \)

### Part (c):
The resultant vector \( \mathbf{R} \) in unit vector notation is:
\[
\mathbf{R} = (-2 \, \text{m})\mathbf{i} + (2 \, \text{m})\mathbf{j}
\]

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Would you like further clarification or more details on any part of the solution?

### Related Questions:
1. How do we apply the tip-to-tail method for adding vectors?
2. What are the geometric interpretations of vector components?
3. Can we use another method (like the parallelogram method) for vector addition?
4. What is the significance of unit-vector notation in vector calculations?
5. How can you find the magnitude of the resultant vector from its components?

### Tip:
When adding vectors graphically, ensure to always align the vectors tip-to-tail and check the resulting direction and magnitude visually before calculating numerically.

In addition to the vectors shown in the figure, we can define another vector. The vector D is specified in unit-vector notation as D = (2 m)x + (-1 m)y. The task is to find the resultant vector obtained by adding vector A and vector D using the tip-to-tail method. Fill in the x and y components of the two vectors and calculate the resultant vector R = A + D.

Learn how to solve vector addition using the tip-to-tail method for vectors A and D. This step-by-step guide helps calculate the x and y components of the resultant vector in unit-vector notation. Suitable for Grades 10-12.

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