Why does a light beam in a moving light clock follows a V shaped path? If I am standing on a platform moving along the X-axis and I throw a basketball vertically upward along the z-axis, the ball also travels along the x-axis. Does the same apply to a photon? According to a special relativity postulate  speed of light is not affected by the velocity of the inertial frame or the velocity of the emitter. If this postulate is true, shouldn't a photon fired vertically as above should travel only along the z-axis but not along the x-axis?
 A: We have three events here:


*

*photon emitted

*photon hits the mirror

*photon comes back to emitter


We know where and when each of these events happened from the point of view of a traveler on a spaceship. He knows that events 1 and 3 happened in the same place, event 3 happened a little bit later than the event 1. Because the emitter is not moving.
From the point of view of the man on Earth events 1 and 3 happened in different places (and, of course, in different moments of time). Let's say coordinates of event 1 is $x_1=0, y_1=0$, coordinate of event 3 is $x_3=a, y_3=0$. Event 2 happened somewhere in between: $x_2 = a/2, y_2=b$. So what was the path of the photon? From event 1 to event 2 to event 3. Photon was not interacting with anything between these events, so it was moving along straight lines. We have a V-shaped path.
It's a simple reasoning and there is nothing relative-theory-specific in it so far. Same reasoning as in a situation with a ball bouncing from the wall on a moving platform.
Difference from classic mechanics is that the speed of photon, traveling along the V shaped path (in the frame of reference of Earth) is the same as speed of the same photon in the frame of reference of the space ship. Both observers claim that the photon has same speed $c$, but they also claim that the distance it traveled between events 1 and 3 is different.
If this postulate is true, shouldn't a photon fired vertically 
as above should travel only along the z-axis but not along the x-axis?

No. Relativity theory only says that the speed of photon must be the same. Direction can be different.
A: If I am in a frame where I watch you go by on the platform, the photon certainly does move along both the $\hat{\mathbf{x}}$ and $\hat{\mathbf{z}}$ axes, and as you mentioned it must travel at speed $c$ in both of our frames. At first glance this might seem inconsistent, because the path length of the photon in your frame appears shorter than the one in my frame. 
What this ends up requiring is for your clock to run at a different rate than mine, a phenomenon referred to as time dilation. For example, imagine the photon reflecting up and down between two mirrors on your moving platform. If you measure a time $\Delta t$ for it to travel between the mirrors, and (watching as you go by) I measure that it takes a time $\Delta t'$, you can find that these time intervals will be related by:
$$
\Delta t' ~=~ \frac{\Delta t}{\sqrt{1 - \frac{v^2}{c^2}}}
$$
where $v$ is the relative speed between our frames. Note that $\Delta t' \geq \Delta t$ so that I always view your clock as running slower. This is what must occur if we demand the speed of light is the same in all reference frames.
A: Now, Imagine this. If light doesn't move in the direction along the spacecrafts velocity, it would not keep reflecting since the spacecraft will have moved a little in that time. The light appears to be reflecting vertically for the person in the spacecraft since both the person and the photon have the same velocity component in the direction of the spacecrafts motion. Yes, this is exactly what happens for a ball.


*

*And yes it is also TRUE that everyone agrees that the SPEED (not velocity) of light is exactly 299792458m/s. Why?


Speed is defined as the distance traveled per unit time. Speed= Distance divided by time. So, since the speed of light is the same in all reference frames, the ratio of the distance traveled and the time taken must be the same.(Not the distance itself).
So since for a stationary observer, the path of light appears to be travelling in a jigsaw pattern, the distance light travels is longer. To compensate this and to maintain lights invariant speed, time runs faster for the stationary observer compared to the observer in the spacecraft.(Since for the observer in the spacecraft, the light only reflect vertically. Hence a smaller distance divided by a smaller time period.). But in the stationary observers case, since time runs faster, a larger distance divided by a larger time period. Hence the ratio remains the same.
-Time slows down just enough so that the Ratio of the distance traveled and time taken remains the same for all observers. 
-This is why we equate the speeds.
--To sum it all up, the Ratio of the distance traveled to the time taken must be and is the same for all observers. Not the distance nor the direction.  Since speed= distance/time.  To maintain this time slows down for the observer in the spacecraft.
A: In reality the basketball was not thrown vertically or horizontally. It was thrown at an angle. The photon also travels at some angle, in a straight line from the sources location at the time it was emitted. 
