Question

Let \(A, B,\) and \(C\) be vertices of a triangle. Find \(\overrightarrow{A B}+\overrightarrow{B C}+\overrightarrow{C A}\)

Short answer

The sum of the vectors \(\overrightarrow{AB} + \overrightarrow{BC} + \overrightarrow{CA}\) is \(0\).

Step-by-step solution

Calculate the Vector AB

The vector AB is represented as \( \overrightarrow{AB} = \overrightarrow{OB} - \overrightarrow{OA} \), where O is the origin.

Calculate the Vector BC

The vector BC is represented as \( \overrightarrow{BC} = \overrightarrow{OC} - \overrightarrow{OB} \), where O is the origin.

Calculate the Vector CA

The vector CA is represented as \( \overrightarrow{CA} = \overrightarrow{OA} - \overrightarrow{OC} \), where O is the origin.

Sum up the Vector AB, BC, and CA

Add up the vectors \( \overrightarrow{AB}, \overrightarrow{BC} \) and \( \overrightarrow{CA} \) to find \( \overrightarrow{AB} + \overrightarrow{BC} + \overrightarrow{CA} \). As a result we get \( (\overrightarrow{OB} - \overrightarrow{OA}) + (\overrightarrow{OC} - \overrightarrow{OB}) + (\overrightarrow{OA} - \overrightarrow{OC}) \).

Simplify the Equation

Simplify the equation from Step 4. Which gives us \( \overrightarrow{OB} - \overrightarrow{OA} + \overrightarrow{OC} - \overrightarrow{OB} + \overrightarrow{OA} - \overrightarrow{OC} \). After simplifying by cancelling equal terms the result becomes \( 0 \).