To express the vector $\vec{C}$ in terms of the unit vectors $\hat{i}$ and $\hat{j}$, we first need to calculate $\vec{C} = 3.00\vec{A} - 4.00\vec{B}$ using the given information.

Given:

1. Vector $\vec{A}$: $\vec{A} \approx 1.23\hat{i} + 3.38\hat{j} \, \text{meters}$

2. Vector $\vec{B}$:

   • Magnitude of $\vec{B} = 2.40$ meters.
   • Angle with the negative x-axis is $30.0^\circ$.
   • Since $\vec{B}$ is in the third quadrant, the x-component will be negative, and the y-component will also be negative.

   The components of $\vec{B}$ are: $B_x = -B \cos(30.0^\circ) = -2.40 \cos(30.0^\circ)$ $B_y = -B \sin(30.0^\circ) = -2.40 \sin(30.0^\circ)$

   Let's calculate: $B_x = -2.40 \times 0.866 \approx -2.08 \, \text{meters}$ $B_y = -2.40 \times 0.500 \approx -1.20 \, \text{meters}$

   Therefore: $\vec{B} \approx -2.08\hat{i} - 1.20\hat{j} \, \text{meters}$

Now, calculate $\vec{C} = 3.00\vec{A} - 4.00\vec{B}$:

$\vec{C} = 3.00(1.23\hat{i} + 3.38\hat{j}) - 4.00(-2.08\hat{i} - 1.20\hat{j})$

Expand and combine like terms:

$\vec{C} = (3.69\hat{i} + 10.14\hat{j}) + (8.32\hat{i} + 4.80\hat{j})$

Combine the components:

$\vec{C} = (3.69 + 8.32)\hat{i} + (10.14 + 4.80)\hat{j}$

Simplifying:

$\vec{C} = 12.01\hat{i} + 14.94\hat{j} \, \text{meters}$

Final Answer:

$\vec{C} \approx 12.01\hat{i} + 14.94\hat{j} \, \text{meters}$

Do you want details or have any questions?

Here are 5 related questions:

1. How do you multiply a vector by a scalar?
2. What are the properties of vector addition and subtraction?
3. How can you find the magnitude of vector $\vec{C}$ after combining $\vec{A}$ and $\vec{B}$?
4. How do you determine the direction of vector $\vec{C}$ based on its components?
5. What happens if vectors $\vec{A}$ and $\vec{B}$ are perpendicular to each other?

Tip: When dealing with vectors, always be mindful of the signs of the components, especially when the vectors are in different quadrants.