# How are the angles equal?

At the back of my mind I know they should be equal, but mathematically, how are the two $\Delta \phi$ angles equal?

The only explanation present in the text is that, "both velocities are perpendicular to the radii vectors," but I don't see how that makes them equal.

Also how will you make those two triangles in the 2 diagrams similar? Any images to support the explanation would be appreciated.

• Just look at how the velocity vectors are perfectly orthogonal to the radii. – ACuriousMind Jul 1 '14 at 17:35
• Yes I know and as i said, at the back of my mind I understand they should be equal, just that a diagram or such with an explanation mathematically could make it better for me :) – Total Anime Immersion Jul 1 '14 at 17:37
• Look at picture A. You can move both vectors $v_1$ and $v_2$ to the origin O, without changing their orientation. Then the angle between both is still the same and you will be able to see that both angles of $v_1$,$v_2$ and $r_1$,$r_2$ are the same, by rotating $v_1$ and $v_2$ by 90 degrees. – physicsGuy Jul 1 '14 at 17:55
• That is what makes me feel they are the same, but what I actually need is a diagram to explain it to me mathematically how the angles are equal :) – Total Anime Immersion Jul 1 '14 at 18:05

Imagine what happens when $\Delta \phi$ keeps increasing to make a full rotation of $360^ \circ$. Then the angle of $P_2$ increases by $360^ \circ$ so that $P_2$ comes back to $P_1$. Also we know that after the full rotation $\vec{v}_1$ must be equal to $\vec{v}_2$ again. Since $\vec{v}_2$ is going around in a circle at the same time $P_2$ does, its angle with $\vec{v}_1$ seems like it should be the same as the angle $P_2$ makes with $P_1$.
More rigourously, the direction of $\vec{v}_1$ is just the direction of $P_1$ rotated by $90^\circ$. Similarly the direction of $\vec{v}_2$ is just the direction of $P_2$ rotated by $90^\circ$. Then since the difference in angle between $P_1$ and $P_2$ is $\Delta \phi$, and $\vec{v}_1$ and $\vec{v}_2$ are essentially rigidly rotated copies of $P_1$ and $P_2$, the angle between $\vec{v}_1$ and $\vec{v}_2$ must also be $\Delta \phi$.