Use unit vectors to express the vector C⃗, where C⃗=3.00A⃗−4.00B⃗.


**Express your answer in terms of the unit vectors** i^ **and** j^**. Use the 'unit vector' button to denote unit vectors in your answer. Express the coefficients in meters**

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.

Learn how to express vector C = 3.00A - 4.00B in terms of unit vectors i and j. This problem involves vector addition, scalar multiplication, and unit vector representation. Suitable for Grades 10-12.

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