To determine vector B⃗ in component form, we use the given vector equation:

$A⃗ + B⃗ + C⃗ = \mathbf{1\hat{j}}$

This equation tells us that the sum of the three vectors results in the unit vector 1 in the j-direction.

Step 1: Express Vectors in Component Form

Let:
$A⃗ = A_x \hat{i} + A_y \hat{j}$ $B⃗ = B_x \hat{i} + B_y \hat{j}$ $C⃗ = C_x \hat{i} + C_y \hat{j}$

The equation then expands as:

$(A_x + B_x + C_x) \hat{i} + (A_y + B_y + C_y) \hat{j} = 0\hat{i} + 1\hat{j}$

Step 2: Solve for $B_x$ and $B_y$

This results in two separate component equations:

1. $A_x + B_x + C_x = 0$ (for the i component)
2. $A_y + B_y + C_y = 1$ (for the j component)

If you provide the values of $A⃗$ and $C⃗$, we can solve for B⃗ explicitly.

Would you like to upload the figure so I can extract the values of $A⃗$ and $C⃗$?

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Related Questions

1. How do you find the resultant of three vectors graphically?
2. What is the significance of unit vectors in vector addition?
3. How do you decompose a vector into its components?
4. What is the method for solving vector equations algebraically?
5. How does the parallelogram rule relate to vector addition?

Tip: Always check your units when working with vector components to avoid calculation errors!