If A, B, and C are the vertices of a triangle ABC, what is the value of vector AB + BC + CA?

To find the value of the vector sum AB + BC + CA, we can start by expressing each vector in terms of the position vectors of the points A, B, and C.

Define the position vectors as follows:

• Let vector A = 𝑟_A
• Let vector B = 𝑟_B
• Let vector C = 𝑟_C

Now, we can express each side of the triangle as:

• Vector AB = 𝑟} – 𝑟}
• Vector BC = 𝑟} – 𝑟}
• Vector CA = 𝑟} – 𝑟}

Now, when we sum these vectors:

AB + BC + CA = (𝑟} - 𝑟}) + (𝑟} - 𝑟}) + (𝑟} - 𝑟})

When we simplify this expression: the 𝑟} and −𝑟} cancel out, as do 𝑟} and −𝑟}, and 𝑟} and −𝑟}.

This shows that:

AB + BC + CA = 0

Hence, the value of the vector sum AB + BC + CA is 0. This indicates that the vectors form a closed triangle in vector space, returning to the starting point when summed together.