Given the vectors A⃗ = 2.00ˆi + 3.00ˆj and B⃗ = 3.00ˆi − 5.00ˆj,

• Draw the vector sum C⃗ = A⃗ + B⃗ on the x − y plane.

• Draw the vector difference D⃗ = A⃗ + B⃗ on the x − y plane.

• Calculate C⃗ and D⃗ with the components.

• Calculate A⃗ · B⃗ .

Let's work through each part of the problem step by step.

### 1. **Draw the vector sum** \( \vec{C} = \vec{A} + \vec{B} \) **on the x-y plane.**

- The vectors are given as:
  - \( \vec{A} = 2.00 \hat{i} + 3.00 \hat{j} \)
  - \( \vec{B} = 3.00 \hat{i} - 5.00 \hat{j} \)

- The vector sum \( \vec{C} = \vec{A} + \vec{B} \):
  \[
  \vec{C} = (2.00 \hat{i} + 3.00 \hat{j}) + (3.00 \hat{i} - 5.00 \hat{j}) = (2.00 + 3.00) \hat{i} + (3.00 - 5.00) \hat{j}
  \]
  \[
  \vec{C} = 5.00 \hat{i} - 2.00 \hat{j}
  \]

  The sum vector \( \vec{C} \) is \( 5.00 \hat{i} - 2.00 \hat{j} \), which is a vector pointing 5 units along the positive x-axis and 2 units down along the y-axis.

### 2. **Draw the vector difference** \( \vec{D} = \vec{A} - \vec{B} \) **on the x-y plane.**

- The vector difference \( \vec{D} = \vec{A} - \vec{B} \):
  \[
  \vec{D} = (2.00 \hat{i} + 3.00 \hat{j}) - (3.00 \hat{i} - 5.00 \hat{j}) = (2.00 - 3.00) \hat{i} + (3.00 - (-5.00)) \hat{j}
  \]
  \[
  \vec{D} = -1.00 \hat{i} + 8.00 \hat{j}
  \]

  The difference vector \( \vec{D} \) is \( -1.00 \hat{i} + 8.00 \hat{j} \), which is a vector pointing 1 unit left along the x-axis and 8 units up along the y-axis.

### 3. **Calculate** \( \vec{C} \) **and** \( \vec{D} \) **with the components.**

We already calculated the components:

- \( \vec{C} = 5.00 \hat{i} - 2.00 \hat{j} \)
- \( \vec{D} = -1.00 \hat{i} + 8.00 \hat{j} \)

### 4. **Calculate** \( \vec{A} \cdot \vec{B} \) (dot product).

The dot product of two vectors \( \vec{A} = A_x \hat{i} + A_y \hat{j} \) and \( \vec{B} = B_x \hat{i} + B_y \hat{j} \) is calculated as:

\[
\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y
\]

For the given vectors:

\[
\vec{A} \cdot \vec{B} = (2.00)(3.00) + (3.00)(-5.00)
\]
\[
\vec{A} \cdot \vec{B} = 6.00 - 15.00 = -9.00
\]

So, the dot product \( \vec{A} \cdot \vec{B} = -9.00 \).

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Now that we've computed the necessary values, let me know if you'd like further details or a visual representation!

### Related Questions:
1. How do you graphically represent vector addition and subtraction?
2. What are the real-world applications of dot products in physics?
3. How do vector components help in calculating the magnitude of a vector?
4. What is the significance of the dot product result being negative?
5. Can we use vectors in higher dimensions? How would this change the operations?

### Tip:  
For vector addition, always align vectors tail-to-head graphically to make the process easier to visualize.

Given the vectors A⃗ = 2.00ˆi + 3.00ˆj and B⃗ = 3.00ˆi − 5.00ˆj, • Draw the vector sum C⃗ = A⃗ + B⃗ on the x − y plane. • Draw the vector difference D⃗ = A⃗ − B⃗ on the x − y plane. • Calculate C⃗ and D⃗ with the components. • Calculate A⃗ · B⃗ .

Learn how to perform vector addition, subtraction, and calculate the dot product for vectors A⃗ = 2.00ˆi + 3.00ˆj and B⃗ = 3.00ˆi − 5.00ˆj. This guide covers essential vector operations and is suitable for students in Grades 11-12.

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