To find the direction of vector $\vec{C} = \vec{A} + \vec{B}$, expressed as an angle counterclockwise from the positive $x$-axis, we will follow these steps:

Given Vectors

$\vec{A} = -4.66\hat{i} - 3.02\hat{j}$ $\vec{B} = -2.47\hat{i} - 4.49\hat{j}$

Step 1: Find the Components of $\vec{C}$

The vector $\vec{C}$ is the sum of $\vec{A}$ and $\vec{B}$: $\vec{C} = \vec{A} + \vec{B} = (-4.66 - 2.47)\hat{i} + (-3.02 - 4.49)\hat{j}$ Calculate the components: $C_x = -4.66 - 2.47 = -7.13$ $C_y = -3.02 - 4.49 = -7.51$

Step 2: Calculate the Angle

The direction angle $\theta_C$ can be found using the arctangent function: $\theta_C = \arctan\left(\frac{C_y}{C_x}\right) = \arctan\left(\frac{-7.51}{-7.13}\right)$

Step 3: Adjust for the Correct Quadrant

Since both $C_x$ and $C_y$ are negative, $\vec{C}$ lies in the third quadrant. The arctangent function will give an angle in reference to the negative $x$-axis. To convert this into the standard counterclockwise angle from the positive $x$-axis: $\theta_C = \arctan\left(\frac{7.51}{7.13}\right) + 180^\circ$

Step 4: Calculation

Now let's compute this using the values: $\theta_C = \arctan\left(\frac{7.51}{7.13}\right) + 180^\circ$

I'll perform this calculation to find the final angle.The direction of the vector $\vec{C} = \vec{A} + \vec{B}$, expressed as an angle counterclockwise from the positive $x$-axis, is approximately $226.49^\circ$.

Do you have any questions or would you like further details?

Here are 5 related questions to explore further:

1. How would you find the angle if the vector lies in a different quadrant?
2. How can you express the direction in radians instead of degrees?
3. What is the significance of vector direction in physics problems?
4. How do vector addition and subtraction affect the resulting direction?
5. What if the vectors had positive components; how would that change the calculations?

Tip: Always pay attention to the signs of the vector components to identify the correct quadrant for the resulting angle.