We are tasked with finding the vector $\vec{C}$ such that:

$\vec{A} + \vec{B} + \vec{C} = \vec{0}$

To do this, we'll first determine the components of vectors $\vec{A}$ and $\vec{B}$, and then solve for $\vec{C}$.

Step 1: Resolve vector $\vec{A}$

From the image, vector $\vec{A}$ has a magnitude of 4 m and is at an angle of 60° above the positive x-axis. To find the components of $\vec{A}$:

$A_x = A \cdot \cos(60^\circ) = 4 \cdot \cos(60^\circ) = 4 \cdot 0.5 = 2 \, \text{m}$ $A_y = A \cdot \sin(60^\circ) = 4 \cdot \sin(60^\circ) = 4 \cdot \frac{\sqrt{3}}{2} = 2\sqrt{3} \, \text{m} \approx 3.46 \, \text{m}$

Thus, the components of $\vec{A}$ are:

$\vec{A} = 2 \hat{i} + 3.46 \hat{j}$

Step 2: Resolve vector $\vec{B}$

Vector $\vec{B}$ has a magnitude of 3 m and is at an angle of 20° below the positive x-axis. To find the components of $\vec{B}$:

$B_x = B \cdot \cos(20^\circ) = 3 \cdot \cos(20^\circ) \approx 3 \cdot 0.94 = 2.82 \, \text{m}$ $B_y = B \cdot \sin(20^\circ) = 3 \cdot \sin(20^\circ) \approx 3 \cdot 0.34 = 1.03 \, \text{m}$

Since the y-component of $\vec{B}$ is pointing downward (negative y-direction), we take:

$B_y = -1.03 \, \text{m}$

Thus, the components of $\vec{B}$ are:

$\vec{B} = 2.82 \hat{i} - 1.03 \hat{j}$

Step 3: Solve for vector $\vec{C}$

Using the equation $\vec{A} + \vec{B} + \vec{C} = \vec{0}$, we can solve for $\vec{C}$:

$\vec{C} = -(\vec{A} + \vec{B})$

First, sum the components of $\vec{A}$ and $\vec{B}$:

$\vec{A} + \vec{B} = (2 + 2.82)\hat{i} + (3.46 - 1.03)\hat{j}$ $\vec{A} + \vec{B} = 4.82 \hat{i} + 2.43 \hat{j}$

Now, the components of $\vec{C}$ are:

$\vec{C} = -(4.82 \hat{i} + 2.43 \hat{j})$ $\vec{C} = -4.82 \hat{i} - 2.43 \hat{j}$

Final Answer:

The vector $\vec{C}$ in component form is:

$\vec{C} = -4.82 \hat{i} - 2.43 \hat{j} \, \text{m}$

Would you like further details or explanations?

Here are five related questions to expand on this topic:

1. How would the solution change if vector $\vec{A}$ had a different angle?
2. How can you calculate the magnitude of vector $\vec{C}$?
3. What is the geometric interpretation of the sum of vectors $\vec{A} + \vec{B} + \vec{C} = 0$?
4. How do you apply vector addition in 3D space?
5. What are the alternative methods for resolving vectors into components?

Tip: Always pay attention to the direction (sign) of vector components when resolving them along axes!