Question

points c, a, and b are collinear. which of the following options is another name for \\(\overline{ca}\\)?

Explanation:

To solve this, we analyze the collinearity of points \( C \), \( A \), and \( B \) (from the diagram, they lie on a straight line). A ray is named by its endpoint first, then another point on the ray. For \( \overrightarrow{CA} \), the endpoint is \( C \), and it passes through \( A \). But since \( C \), \( A \), \( B \) are collinear, \( \overrightarrow{CB} \) (if \( B \) is on the same line from \( C \) through \( A \)) or \( \overrightarrow{CA} \) can also be named as \( \overrightarrow{CB} \) (assuming the diagram shows \( A \) between \( C \) and \( B \) or vice - versa, but from the collinearity, the ray starting at \( C \) and going through \( A \) is the same as the ray starting at \( C \) and going through \( B \) if \( A \) and \( B \) are on the same line from \( C \)). Wait, actually, the key is that for a ray \( \overrightarrow{XY} \), any point \( Z \) on the ray (beyond \( Y \) from \( X \)) can be used to name it as \( \overrightarrow{XZ} \). Since \( C \), \( A \), \( B \) are collinear, \( \overrightarrow{CA} \) is the same as \( \overrightarrow{CB} \) (assuming the direction is from \( C \) through \( A \) to \( B \)). But looking at the options (even though not fully visible, from the context of collinear points \( C \), \( A \), \( B \)), the ray \( \overrightarrow{CA} \) can also be named as \( \overrightarrow{CB} \) (if \( B \) is on the ray \( \overrightarrow{CA} \)) or \( \overrightarrow{AC} \) is not, but \( \overrightarrow{CA} \) and \( \overrightarrow{CB} \) (if \( A \) and \( B \) are on the same line from \( C \)).

Assuming the options are like \( \overrightarrow{CB} \), \( \overrightarrow{AC} \), \( \overrightarrow{AB} \), \( \overrightarrow{BA} \), the correct one is \( \overrightarrow{CB} \) (since \( C \), \( A \), \( B \) are collinear, the ray starting at \( C \) and passing through \( A \) also passes through \( B \), so \( \overrightarrow{CA}=\overrightarrow{CB} \)).

Step1: Recall Ray Naming

A ray \(\overrightarrow{XY}\) is named by endpoint \(X\) and a point \(Y\) on it.

Step2: Use Collinearity

Since \(C\), \(A\), \(B\) are collinear, the ray starting at \(C\) through \(A\) also passes through \(B\). So \(\overrightarrow{CA}\) can be named as \(\overrightarrow{CB}\) (as \(B\) is on the ray from \(C\) through \(A\)).

Answer:

\(\boldsymbol{\overrightarrow{CB}}\) (assuming the collinear points and ray naming convention)