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The resultant of p vector and q vector is r vector. When the direction of q vector is reverse then the resultant is s vector . Prove that sq of r+sq of s =2(sq of p+sq of q)

The resultant of p vector and q vector is r vector. When the direction of q vector is reverse then the resultant is s vector . Prove that sq of r+sq of s =2(sq of p+sq of q)

Grade:11

3 Answers

Arun
25763 Points
4 years ago
Please draw a parallelogram AB CD.  Let CD be parallel to AB. Let BC be parallel to AD.  Let vector P be represented by AB. Let Q be represented by AD.  Now the resultant force of P & Q is given by parallelo gram law as
         R =  square root [ P²+ Q² + 2 P Q cos A ]  = This is diagonal AC
If the angle is reversed, That is, reverse vector AD or Q. Now draw parallelogram AEFB so that AE = - Q = - AD.   FB = EA.  Now the diagonal  AF will be parallel to diagonal BD.  The angle at A in AEFB is  180 - A.  SO its cosine is - Cos A.
 S = squareroot [ P² + Q² - 2 P Q cos A ]
 So R² + Q² = 2 [P² + Q² ]             
as the other terms cancel.
This is also a relation between sides of a parallelogram and diagonals
Avika
15 Points
2 years ago
R=
Aditya Gupta
2080 Points
2 years ago
the best method to solve this is as follows:
p+q= r 
and p – q= s
taking magnitude and squaring
p^2 + q^2 + 2p.q= r^2
p^2 + q^2 – 2p.q= s^2
adding
2(p^2 + q^2)= r^2+s^2.
 

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