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.