Any two-dimensional vector, such as momentum, can be resolved into horizontal ( x ) and vertical ( y ) components. 

The diagram to the right shows a vector,  «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mover»«mi»p«/mi»«mo»§#8594;«/mo»«/mover»«/math» , resolved into its  x  and  y  components.

 

Using the above diagram t and the trigonometric identities for sine, cosine, and tangent, answer the following Self-Check questions.   

What is the expression that gives px as a function of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mover»«mi»p«/mi»«mo»§#8594;«/mo»«/mover»«/math»  and θ ?

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»p«/mi»«mi»x«/mi»«/msub»«mo»=«/mo»«mover»«mi»p«/mi»«mo»§#8594;«/mo»«/mover»«mi»cos«/mi»«mi»§#952;«/mi»«/math»


What is the expression that gives py as a function of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mover»«mi»p«/mi»«mo»§#8594;«/mo»«/mover»«/math»  and θ ?

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»p«/mi»«mi»y«/mi»«/msub»«mo»=«/mo»«mover»«mi»p«/mi»«mo»§#8594;«/mo»«/mover»«mi»sin«/mi»«mi»§#952;«/mi»«/math»


Write a general expression for «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mover»«mi»p«/mi»«mo»§#8594;«/mo»«/mover»«/math» , if both px and py are known.

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mover»«mi»p«/mi»«mo»§#8594;«/mo»«/mover»«mo»=«/mo»«msqrt»«msubsup»«mi»p«/mi»«mi»x«/mi»«mn»2«/mn»«/msubsup»«mo»§#160;«/mo»«mo»+«/mo»«mo»§#160;«/mo»«msubsup»«mi»p«/mi»«mi»y«/mi»«mn»2«/mn»«/msubsup»«/msqrt»«/math»


If both px and py are known, what is the general expression for θ?

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»§#952;«/mi»«mo»=«/mo»«msup»«mi»tan«/mi»«mrow»«mo»-«/mo»«mn»1«/mn»«/mrow»«/msup»«mfenced open=¨[¨ close=¨]¨»«mfrac»«msub»«mi»p«/mi»«mi»y«/mi»«/msub»«msub»«mi»p«/mi»«mi»x«/mi»«/msub»«/mfrac»«/mfenced»«/math»