Given a vector drawn on a two dimensional coordinate axis, how does one calculate the componant of the vector projected along the x-axis? 

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mwmovr40 eNotes educator| Certified Educator

A vector is a physical quantity which has not only a size, but also a direction.  For example: to walk 30 meters North-East is a vector quantity (displacement) because it includes a size (30 m) and a direction (North-East).  Quantities, such as mass, which do not require a direction are called scalars.

We can represent a vector by making a scale drawing on a coordinate plane.  When it is possible to do so, we pick the origin of the coordinate plane to be where the vector begins.  The vector is then drawn using an arrow that has a length proportional to the size of the vector and is pointing in the same direction as the vector. 

For example: if the above vector is drawn on standard 1/4" graph paper and each quarter of an inch represents 1 meter, then the length of the arrow on would be 7.5 inches and would be pointing 45 degrees up from the x-axis.

Any vector can be represented by a pair of horizontal (x-axis) and vertical (y-axis) components.  The two component represents the straight line distance in the horizontal and vertical directions one would have to measure from the beginning of the vector to reach the end of the vector.  When these two vectors are drawn on the same coordinate axis as the vector they create the two legs of a right triangle and the vector creates the hypotenuse of the right triangle.

Using simple right-angle trigonometry we can calculate the length of each component if we know the original lenght of the vector and the angle it makes to the x-axis.  From trig we know

`costheta = x/(hypotenuse)`   

`sintheta = y/(hypotenuse)`

Recalling that the hypotenuse is also the length of the original vector, the x-component and y-components would be

`x = Vcostheta`  

 `y = Vsintheta`

 Using the example above the x-axis component would be

`x = 30m* cos(45) = 21.2 m`

and the y-axis component would be

`y = 30m*sin(45) = 21.2m`