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the resultant of two vectors P and Q is R. if Q is doubled, the new resultant is perpendicular to P. Then R equals (a) P (b) (P+Q) ( c) Q (d) (P-Q) please explain how to solve

the resultant of two vectors P and Q is R. if Q is doubled, the new resultant is perpendicular to P. Then R equals
(a) P
(b) (P+Q)
( c) Q
(d) (P-Q)
 please explain how to solve

Grade:11

4 Answers

Saurabh Kumar
askIITians Faculty 2400 Points
8 years ago
245-136_HGHG.PNG
Surya singh
36 Points
6 years ago
For first scene when resultant of P and Q is R
Vector R=Vector P +Vector Q
R=P2 +Q2+2PQCos\Theta….................equation 1
Then after doubling the magnitude of Q the new resultant is perpendicular to P..That means dot product of P and new resultant(R’) will be zero.
R’ * P=0...................equation 2
(where R’=P+2Q)
by putting value to equation 2
we get Cos\Theta = -P/2Q............equation 3
So from equation 1 and 3,we get 
R=Q
 
Kushagra Madhukar
askIITians Faculty 628 Points
3 years ago
Hello student
 
By law of vector addition, the resultant vector can be given as
R = P + Q
 
Now when Q is doubled the new resultant vector can be given as
R’ = P + 2Q
since, R’ is perpendicular to P
hence there dot product should be zero
or, R’ . P = 0
or, ( P + 2Q ).P = |P|2 + 2Q.P = 0
or, 2P.Q = -|P|2
 
Now,  |R|2 = |P|2 + |Q|2 + 2P.Q
                 =  |P|2 + |Q|2 – |P|2
                 =  |Q|2
or, R = Q
 
Hope it helps
Regards
Kushagra
Yash Chourasiya
askIITians Faculty 256 Points
3 years ago
Dear Student

643-1699_Untitled.png
Let the angle between two vectorsP and Qbeαand their resultant isR

So we can write

R2 = P2 + Q2 + 2PQcosα......[1]

When Q is doubled then let the resultant vector beR1, So we can write

R12= P2 + 4Q2 + 4PQcosα......[2]

Again by the given conditionR1is perpendicular toP

So4Q2 = P2 + R12......[3]

Combining [2] and [3] we get

R12 = P2 + P2 + R12 + 4PQcosα

⇒ 2PQcosα = −P2......[4]

combining [1] and [4] we get

R2 = P2 + Q2 − P2

⇒ R = Q

I hope this answer will help you.
Thanks & Regards
Yash Chourasiya

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