the resultant of P and Q is R if Q is double then R is also double and when Q is reverse then R is again double. find the ratio P:Q:R
Answers
Answer:
P:Q:R ::: √2 : √3 : √2
Explanation:
the resultant of P and Q is R if Q is double then R is also double and when Q is reverse then R is again double. find the ratio P:Q:R
Let say P & Q have angle α
R² = P² + Q² + 2PQCosα
=>2PCosα = R² - P² -Q²
if Q is double then R is also double
=> (2R)² = P² + (2Q)² + 2P(2Q)Cosα
=> 4R² = P² + 4Q² + 4PQCosα
=> 4R² = P² + 4Q² + 2(2PQCosα)
=> 4R² = P² + 4Q² + 2(R² - P² -Q²)
=> 4R² = P² + 4Q² + 2R² - 2P² -2Q²
=> 2R² = 2Q² - P²
=> 2Q² = 2R² + P²
when Q is reverse then R is again double
=> (2R)² = P² + (Q)² - 2P(Q)Cosα (as Cos(180 + α) = -Cosα)
=> 4R² = P² + Q² -2PQCosα
=> 2PQCosα = P² + Q² - 4R²
P² + Q² - 4R² = R² - P² -Q²
=> 2Q² = 5R² - 2P²
2R² + P² = 5R² - 2P²
=> 3P² = 3R²
=> P = R
2Q² = 2R² + R²
=> 2Q² = 3R²
=> Q : R :: √3 : √2
P:Q:R ::: √2 : √3 : √2