Question

points c, a, and b are collinear. which of the following are opposite rays? $overrightarrow{ca}$ and $overrightarrow{ab}$, $overrightarrow{ac}$ and $overrightarrow{ab}$, $overrightarrow{ab}$ and $overrightarrow{ad}$, $overrightarrow{ac}$ and $overrightarrow{ad}$

Explanation:

Step1: Recall the definition of opposite rays

Opposite rays are two rays that share the same endpoint, lie on a straight line (are collinear), and extend in opposite directions.

Step2: Analyze each option

• Option 1 ($\boldsymbol{\overrightarrow{CA}}$ and $\boldsymbol{\overrightarrow{AB}}$): $\overrightarrow{CA}$ has endpoint $C$, and $\overrightarrow{AB}$ has endpoint $A$. Different endpoints → Not opposite rays.
• Option 2 ($\boldsymbol{\overrightarrow{AC}}$ and $\boldsymbol{\overrightarrow{AB}}$):
• Common endpoint: $A$.
• Collinear: Points $C$, $A$, $B$ are collinear (given), so $\overrightarrow{AC}$ (from $A$ through $C$) and $\overrightarrow{AB}$ (from $A$ through $B$) lie on the same straight line.
• Opposite directions: From $A$, $\overrightarrow{AC}$ extends toward $C$ (one direction), and $\overrightarrow{AB}$ extends toward $B$ (the opposite direction along the line $CB$).

This satisfies all conditions for opposite rays.

• Option 3 ($\boldsymbol{\overrightarrow{AB}}$ and $\boldsymbol{\overrightarrow{AD}}$): $\overrightarrow{AB}$ and $\overrightarrow{AD}$ form an angle (not collinear) → Not opposite rays.
• Option 4 ($\boldsymbol{\overrightarrow{AC}}$ and $\boldsymbol{\overrightarrow{AD}}$): $\overrightarrow{AC}$ and $\overrightarrow{AD}$ form an angle (not collinear) → Not opposite rays.

Thus, the correct answer is $\boldsymbol{\overrightarrow{AC}}$ and $\boldsymbol{\overrightarrow{AB}}$.

Answer:

$\overleftrightarrow{CA}$ and $\overleftrightarrow{AB}$ (Wait, actually, the correct opposite rays should have the same endpoint and form a straight line. Looking at the diagram, points C, A, B are collinear. So ray $\overrightarrow{CA}$ and ray $\overrightarrow{AB}$? Wait no, opposite rays have the same endpoint and go in opposite directions along a straight line. So the endpoint is A, and the two rays should be $\overrightarrow{AC}$ and $\overrightarrow{AB}$? Wait no, let's recall: Opposite rays are two rays that share the same endpoint and form a straight line (180 degrees). So from point A, ray going to C and ray going to B? Wait the options: Let's check the options. The options are:

1. $\overrightarrow{CA}$ and $\overrightarrow{AB}$

1. $\overrightarrow{AC}$ and $\overrightarrow{AB}$

1. $\overrightarrow{AB}$ and $\overrightarrow{AD}$

1. $\overrightarrow{AC}$ and $\overrightarrow{AD}$

Wait, first, opposite rays must have the same endpoint and be collinear (lie on a straight line) and go in opposite directions. So endpoint A: rays $\overrightarrow{AC}$ and $\overrightarrow{AB}$? Wait no, $\overrightarrow{AC}$ goes from A to C, $\overrightarrow{AB}$ goes from A to B. But C, A, B are collinear, so A is between C and B. So $\overrightarrow{AC}$ (from A to C) and $\overrightarrow{AB}$ (from A to B) – wait, no, opposite rays would be, for example, $\overrightarrow{AC}$ and $\overrightarrow{AB}$? Wait no, if C---A---B, then the ray starting at A through C is $\overrightarrow{AC}$ (wait, no, ray $\overrightarrow{AC}$ starts at A and goes to C? No, ray notation: $\overrightarrow{AC}$ starts at A and goes through C, so direction is A to C. $\overrightarrow{AB}$ starts at A and goes through B. But since C, A, B are collinear, with A between C and B, then $\overrightarrow{AC}$ (A to C) and $\overrightarrow{AB}$ (A to B) are in opposite directions? Wait no, $\overrightarrow{CA}$ starts at C and goes to A, $\overrightarrow{AB}$ starts at A and goes to B – no, same endpoint? No, $\overrightarrow{CA}$ has endpoint C, $\overrightarrow{AB}$ has endpoint A. So that's not same endpoint. Wait, the correct opposite rays must have the same endpoint. So let's re-express:

Opposite rays: share the same endpoint, and their union is a straight line (so they are collinear and in opposite directions).

So endpoint A:

• Ray $\overrightarrow{AC}$: starts at A, goes through C (direction A→C)

• Ray $\overrightarrow{AB}$: starts at A, goes through B (direction A→B)

But since C---A---B, then A is between C and B, so $\overrightarrow{AC}$ (A to C) and $\overrightarrow{AB}$ (A to B) are in opposite directions? Wait, no, if you are at A, going to C is one direction, going to B is the opposite direction along the line CB. So yes, $\overrightarrow{AC}$ and $\overrightarrow{AB}$? Wait but the first option is $\overrightarrow{CA}$ and $\overrightarrow{AB}$ – $\overrightarrow{CA}$ starts at C, endpoint C, direction C→A. $\overrightarrow{AB}$ starts at A, endpoint A, direction A→B. Different endpoints, so not opposite rays.

Second option: $\overrightarrow{AC}$ (endpoint A, direction A→C) and $\overrightarrow{AB}$ (endpoint A, direction A→B). Wait, but if C---A---B, then A→C and A→B are opposite directions? Wait no, A→C is towards C, A→B is towards B, which are opposite along the line. So yes, they are opposite rays. Wait but let's check the diagram again. The diagram shows:

• A is a point, with C above (wait, no, the arrows: the line with C and B has arrows, so C---A---B, with arrows pointing up (C) and down (B), so the line is straight, C, A, B collinear. Then AD is a ray from A to D, at an angle.

So opposite rays must have same endpoint, collinear, opposite directions.

So endpoint A:

• Ray $\overrightarrow{AC}$: starts at A, goes through C (direction A to C)

• Ray $\overrightarrow{AB}$: starts at A, goes through B (direction A to B)

These two rays have the same endpoint A, lie on the straight line CB, and go in opposite directions (since A is between C and B). So they are opposite rays. Wait but the second option is $\overrightarrow{AC}$ and $\overrightarrow{AB}$? Wait the options are:

1. $\overrightarrow{CA}$ and $\overrightarrow{AB}$

1. $\overrightarrow{AC}$ and $\overrightarrow{AB}$

1. $\overrightarrow{AB}$ and $\overrightarrow{AD}$

1. $\overrightarrow{AC}$ and $\overrightarrow{AD}$

Wait, no, maybe I misread the options. Let me check again. The user's image:

Options:

• $\overrightarrow{CA}$ and $\overrightarrow{AB}$

• $\overrightarrow{AC}$ and $\overrightarrow{AB}$

• $\overrightarrow{AB}$ and $\overrightarrow{AD}$

• $\overrightarrow{AC}$ and $\overrightarrow{AD}$

Wait, no, the first option is $\overrightarrow{CA}$ (endpoint C, direction C to A) and $\overrightarrow{AB}$ (endpoint A, direction A to B) – different endpoints, so not opposite.

Second option: $\overrightarrow{AC}$ (endpoint A, direction A to C) and $\overrightarrow{AB}$ (endpoint A, direction A to B) – same endpoint, collinear (since C, A, B are collinear), and opposite directions (A to C and A to B are opposite along the line). So that's opposite rays. Wait but let's confirm the definition: Opposite rays are two rays that have a common endpoint and form a straight line (i.e., their union is a straight line, so they are collinear and in opposite directions). So yes, $\overrightarrow{AC}$ and $\overrightarrow{AB}$: common endpoint A, collinear (C, A, B are collinear), and opposite directions (since A is between C and B, so from A, going to C is one direction, going to B is the opposite). So the correct answer is the second option: $\overrightarrow{AC}$ and $\overrightarrow{AB}$? Wait but the user's options: let me check the original image again.

Wait the user's image:

The options are:

1. $\overleftrightarrow{CA}$ and $\overleftrightarrow{AB}$ – no, those are line segments or lines? Wait, the notation: $\overrightarrow{CA}$ is a ray, $\overrightarrow{AB}$ is a ray.

Wait, maybe I made a mistake. Let's re-express:

Ray $\overrightarrow{CA}$: starts at C, goes through A.

Ray $\overrightarrow{AB}$: starts at A, goes through B.

Different endpoints (C vs A), so not opposite.

Ray $\overrightarrow{AC}$: starts at A, goes through C.

Ray $\overrightarrow{AB}$: starts at A, goes through B.

Same endpoint (A), collinear (C, A, B are collinear), and opposite directions (since A is between C and B, so from A, going to C is towards C, going to B is towards B, which are opposite along the line). So yes, these are opposite rays.

So the correct answer is the second option: $\overrightarrow{AC}$ and $\overrightarrow{AB}$? Wait but the user's options: let me check the original problem again.

Wait the problem says "Which of the following are opposite rays?"

Options:

1. $\overrightarrow{CA}$ and $\overrightarrow{AB}$

1. $\overrightarrow{AC}$ and $\overrightarrow{AB}$

1. $\overrightarrow{AB}$ and $\overrightarrow{AD}$

1. $\overrightarrow{AC}$ and $\overrightarrow{AD}$

Wait, no, maybe the second option is $\overrightarrow{AC}$ and $\overrightarrow{AB}$? Wait, but in the diagram, C, A, B are collinear, so the line is C---A---B. So the ray from A through C is $\overrightarrow{AC}$, and the ray from A through B is $\overrightarrow{AB}$. These two rays have the same endpoint A, lie on a straight line, and go in opposite directions (since A is between C and B). Therefore, they are opposite rays. So the correct answer is the second option: $\overrightarrow{AC}$ and $\overrightarrow{AB}$? Wait but the user's options: maybe I misread the second option. Let me check again.

Wait the user's image:

The second option is $\overrightarrow{AC}$ and $\overrightarrow{AB}$? Wait, the text in the image:

First option: $\overrightarrow{CA}$ and $\overrightarrow{AB}$

Second option: $\overrightarrow{AC}$ and $\overrightarrow{AB}$

Third option: $\overrightarrow{AB}$ and $\overrightarrow{AD}$

Fourth option: $\overrightarrow{AC}$ and $\overrightarrow{AD}$

Yes. So the correct answer is the second option: $\overrightarrow{AC}$ and $\overrightarrow{AB}$? Wait no, wait: opposite rays must be in opposite directions. So if C---A---B, then the ray starting at A and going to C is $\overrightarrow{AC}$, and the ray starting at A and going to B is $\overrightarrow{AB}$. But since A is between C and B, $\overrightarrow{AC}$ is towards C (left), $\overrightarrow{AB}$ is towards B (right), so they are opposite rays. Yes. So the answer is the second option: $\overrightarrow{AC}$ and $\overrightarrow{AB}$. Wait but let's confirm with the definition.

Definition of opposite rays: Two rays are opposite rays if they have a common endpoint and their union is a straight line (i.e., they are collinear and in opposite directions). So:

• Common endpoint: A

• Collinear: C, A, B are collinear, so $\overrightarrow{AC}$ and $\overrightarrow{AB}$ lie on the same line.

• Opposite directions: From A, $\overrightarrow{AC}$ goes to C (one direction), $\overrightarrow{AB}$ goes to B (the opposite direction along the line).

Therefore, $\overrightarrow{AC}$ and $\overrightarrow{AB}$ are opposite rays. So the correct answer is the second option.