Question

1. in the diagram below, points e, f, and g all lie on the same line. which two geometric objects listed below are not the same? (1) $overrightarrow{eg}$ and $overrightarrow{ge}$ (2) $overrightarrow{ge}$ and $overrightarrow{gf}$ (3) $overrightarrow{ef}$ and $overrightarrow{fg}$ (4) $overrightarrow{ef}$ and $overrightarrow{fe}$

Explanation:

Step1: Recall vector - ray properties

Vectors and rays have directions. $\overrightarrow{AB}$ has its starting - point at $A$ and ending - point at $B$, while $\overrightarrow{BA}$ has its starting - point at $B$ and ending - point at $A$.

Step2: Analyze option (1)

$\overrightarrow{EG}$ starts at $E$ and ends at $G$, and $\overrightarrow{GE}$ starts at $G$ and ends at $E$. They have opposite directions.

Step3: Analyze option (2)

$\overrightarrow{GE}$ starts at $G$ and ends at $E$, $\overrightarrow{GF}$ starts at $G$ and ends at $F$. They are different as they have different end - points.

Step4: Analyze option (3)

$\overrightarrow{EF}$ starts at $E$ and ends at $F$, $\overrightarrow{FG}$ starts at $F$ and ends at $G$. They are different as they have different starting and ending points.

Step5: Analyze option (4)

$\overrightarrow{EF}$ and $\overrightarrow{FE}$ have opposite directions.

Answer:

(1) $\overrightarrow{EG}$ and $\overrightarrow{GE}$; (2) $\overrightarrow{GE}$ and $\overrightarrow{GF}$; (3) $\overrightarrow{EF}$ and $\overrightarrow{FG}$; (4) $\overrightarrow{EF}$ and $\overrightarrow{FE}$ are all pairs of non - identical geometric objects. But if we consider the nature of vectors (direction matters), the most fundamental non - identical pair based on direction is (1) $\overrightarrow{EG}$ and $\overrightarrow{GE}$ since they have opposite directions while the others also involve different starting/ending points in addition to direction differences in some cases. So the answer is (1).