Why Are the Two Angles Labeled Theta Equal?

[xatu]

Homework Statement

In my statics book (13th ed. Hibbeler) I'm reviewing the section about using vector analysis to calculate the moment of a force about a specific axis. I understand the theory fine, but I don't completely understand the figure in the book (pic attached). Specifically, why are the two angles labeled theta equal? The first tangle is between the position vector and the x axis, and the second angle is between the moment and it's y component.

Homework Equations

Trig relations would obviously be of some help.

The Attempt at a Solution

[/B]
Since $r$ is any vector from point O extending to the line of action of the applied force, doesn't that mean we can adjust $r$ and still maintain the same moment $M_O$? Thus $\theta$ between $r$ and the x-axis is changing while $\theta$ between $M_O$ and $M_y$ remains the same?

 

[nasu]

What is Mo?

 

[xatu]

$M_O$ is the moment of the force $F$ about the axis through point O.

 

[nasu]

OK. What is the direction of Mo in respect to the plane determined by F and r? What is the angle between Mo and r?

 

[xatu]

$M_O$ is, by definition, perpendicular to the plane containing $F$ and $r$. The angle between $M_O$ and $r$ is thus 90°.

 

[nasu]

Great. so now you see why the two angles are equal?
The angle between the y-axis and r should give 90 when added to theta, right?

 

[xatu]

Yes, but I don't exactly understand how the angle between the y-axis and $r$, and the angle $\theta$ being complementary to one another tells us anything about the angle between $M_O$ and $M_y$.

I get that the angles adjacent to $\theta$ are equivalent to $90-\theta$, but, to me, that doesn't explain why both $\theta$'s seem to be equal.

Can you please clarify?

 

[nasu]

If we call "alpha" the angle between r and My, it is complementary to both angles labeled "theta".
Or both angles have the same complementary angle (alpha).

 

[xatu]

Thanks a ton, I got it now! Interesting way of looking at things!