The question asks to determine the magnitude and direction of vector C such that the resultant of the three vectors A, B, and C equals zero.

We are given the vectors A and B with their magnitudes and directions based on the image. Let's first calculate these vectors and then determine vector C.

Step 1: Determine vectors A and B from the image.

• Vector A appears to be pointing in the third quadrant (southwest) with coordinates approximately $(-5, 5)$.
• Vector B is in the first quadrant (northeast) with coordinates approximately $(5, 10)$.

Let's write these vectors in component form:

• $\mathbf{A} = (-5, 5)$
• $\mathbf{B} = (5, 10)$

Step 2: Calculate vector C for the resultant to be zero.

Since the resultant of A, B, and C should be zero, we use the vector equation:

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

Rearranging this, we find:

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

Let's compute this:

$\mathbf{C} = - [(-5, 5) + (5, 10)]$ $\mathbf{C} = - [(0, 15)] = (0, -15)$

Step 3: Conclusion

• The magnitude of vector C is $|\mathbf{C}| = 15$ (as it lies entirely in the vertical direction).
• The direction of vector C is downward (along the negative $y$-axis).

Would you like more details or have any other questions?

Here are 5 related questions:

1. How can I calculate the magnitude of a vector in general?
2. What are the vector components in different coordinate systems?
3. How do vector addition and subtraction work graphically?
4. How does one find the direction of a vector using trigonometry?
5. Can you show how to find the resultant of three or more vectors?

Tip: When adding vectors, always break them down into components, as it simplifies the calculations.