Question

For the figure

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

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

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

X

Answer 1

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

Answer 2

The figure displays three vectors $\vec{A}$, $\vec{B}$, and $\vec{C}$ forming a triangle. To determine the correct vector sum, we apply the triangle law of vector addition.

The triangle law states that if two vectors are represented by two sides of a triangle taken in the same order, their resultant is represented by the third side taken in the opposite order.

Let's trace the vectors in the figure:

1. Vector $\vec{A}$ starts from the bottom-left vertex and points to the bottom-right vertex.
2. Vector $\vec{B}$ starts from the head of $\vec{A}$ (bottom-right vertex) and points to the top vertex.
3. Vector $\vec{C}$ starts from the tail of $\vec{A}$ (bottom-left vertex) and points to the head of $\vec{B}$ (top vertex).

According to the triangle law of vector addition: If we add $\vec{A}$ and $\vec{B}$ (which are taken in the same order, head-to-tail), their resultant vector should start from the tail of $\vec{A}$ and end at the head of $\vec{B}$. Observing the figure, the vector $\vec{C}$ precisely matches this description: it starts at the tail of $\vec{A}$ and ends at the head of $\vec{B}$.

Therefore, the correct vector equation is: $\vec{A} + \vec{B} = \vec{C}$

Let's verify the other options:

• $\vec{B} + \vec{C} = \vec{A}$: For this to be true, $\vec{B}$ and $\vec{C}$ must be head-to-tail, and $\vec{A}$ must be the resultant. In the figure, $\vec{B}$ and $\vec{C}$ are not arranged head-to-tail. Specifically, the head of $\vec{B}$ and the head of $\vec{C}$ meet at the top vertex, and the tail of $\vec{B}$ is the head of $\vec{A}$, while the tail of $\vec{C}$ is the tail of $\vec{A}$. This equation is incorrect.
• $\vec{C} + \vec{A} = \vec{B}$: For this to be true, $\vec{C}$ and $\vec{A}$ must be head-to-tail, and $\vec{B}$ must be the resultant. In the figure, the head of $\vec{C}$ is at the top vertex, and the tail of $\vec{A}$ is at the bottom-left vertex. They are not connected head-to-tail. This equation is incorrect.

Thus, only the first statement is correct based on the given figure.