Hey there! We receieved your request
Stay Tuned as we are going to contact you within 1 Hour
One of our academic counsellors will contact you within 1 working day.
Click to Chat
1800-5470-145
+91 7353221155
Use Coupon: CART20 and get 20% off on all online Study Material
Complete Your Registration (Step 2 of 2 )
Sit and relax as our customer representative will contact you within 1 business day
OTP to be sent to Change
Using the formula of vectors -
R^2 = P^2 + Q^2 +2|P||Q|CosΘ ———————-(1)
The given condition is, If Q is doubled, R is doubled means -
(2R)^2 = P^2 + (2Q)^2 +2|P||2Q|CosΘ , on solving gives -
4R^2 = P^2 + 4Q^2 +4|P||Q|CosΘ ————(2)
Another given condition is -
If Q is reversed, R is again doubled, means -
(2R)^2 = P^2 + (-Q)^2 +2|P||-Q|Cos(180-Θ)
IMP Points - 1. Reversing a vector means changing its direction by 180.2. |-Q| = |Q|3. Cos(180-Θ) = -CosΘ
On solving, we get -
4R^2 = P^2 + Q^2 -2|P||Q|CosΘ ———-(3)
Subtract Eqn(3) from Eqn(2),
0 = 3Q^2 + 6|P||Q|CosΘAssuming Q as non-zero vector,-3|Q| = 6|P|CosΘ
|Q| = -2|P|CosΘ ———————————(4)
Squaring both sides, Q^2 = 4P^2(CosΘ)^2 —————(5)
Now subtract Eqn(1) from Eqn(2) which gives -
3R^2 = 3Q^2 + 2|P||Q|CosΘ ————-(6)
Substituting the value obtained from Eqn(5) and Eqn(4) in Eqn (6),
3R^2 = 3[4P^2(CosΘ)^2] + 2|P|[-2|P|CosΘ]CosΘ
3R^2 = 12P^2(CosΘ)^2 - 4P^2(CosΘ)^2
3R^2 = 8P^2(CosΘ)^2 —————- (7)
Substituting the values of R^2 and Q^2, obtained from Eqn(4) & Eqn(5)in Eqn(1) -
[8P^2(CosΘ)^2]3 = P^2 + [4P^2(CosΘ)^2] + 2|P|[-2|P|CosΘ]CosΘ
16P^2(CosΘ)^2 = P^2 + 4P^2(CosΘ)^2 – 4P^2(CosΘ)^2
16P^2(CosΘ)^2 = P^2Assuming P as non-zero vector,
(CosΘ)^2 = 1/16 ———————(8)
We know from Eqn(5), Q^2 = 4P^2(CosΘ)^2& from Eqn(7), R^2 = [8P^2(CosΘ)^2]/3
Put value of (CosΘ)^2 obtained from Eqn(8) in both the above mentioned equations -
Q^2 = 1/4*(P^2)R^2 = 1/6*(P^2)
Now taking the ratio - P^2 : Q^2 : R^2 = P^2 : 1/4*(P^2) : 1/6*(P^2) P^2 : Q^2 : R^2 = 1 : 1/4: 1/6P^2 : Q^2 : R^2 = 12 : 3 : 2 ….. or we can write - P^2 : Q^2 : R^2 = 6 : 1.5 : 1
Get your questions answered by the expert for free
You will get reply from our expert in sometime.
We will notify you when Our expert answers your question. To View your Question
Win Gift vouchers upto Rs 500/-
Register Yourself for a FREE Demo Class by Top IITians & Medical Experts Today !