How Are Triangles Used to Represent Vectors in Physics?

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redroom
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As someone who has just started Physics, when triangles were introduced in my physics it caught me in surprise. If the legs of triangle were heights It would somewhat make more sense to me, but our teacher says their are velocities. I do not understand.

For example :

A pitched ball is hit by a batter at 49 degrees angle. It just clears the outfield fence, 94 m away. the acceleration of gravity is 9.8 m/s squared.

Find the velocity of the ball when it left the bat. Assume the fence is the same height as the pitch.
 
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When we are considering various lengths we are using a good old triangle but when we are considering velocities we are using vector quantities which have components (not legs). A triangle is something tangible velocity vectors are not. The legs of a triangle and the components of a vector are completely different concepts. For example, we can add vectors together but what would it mean to add two triangle? More information on the subject can be found in any matrix theory course.
 
To my knowledge vectors are basically the X and Y axis that the triangle's leg travel on. Is it more to it? Can you please explain to me more why the vectors are needed for this problem and how to use them.
 
redroom said:
To my knowledge vectors are basically the X and Y axis that the triangle's leg travel on. Is it more to it? Can you please explain to me more why the vectors are needed for this problem and how to use them.

Nothing travels on the legs of a triangle. We are describing vectors in a Cartesian coordinate system; vectors have components (analogous to triangle legs) which are easier to describe using the x & y plane.

so,

what is a vector defined to be?
what is a Cartesian coordinate system?