The resultant of P and Q is R. If Q is doubled, R is doubled; when Q is reversed, R is again doubled. Find P: Q: R. (Vectors, Class 11)
Answers
Ratio of P:Q: R = 1 : 0.05 : 0.04
Explanation:
Given: The resultant of P and Q is R. If Q is doubled, R is doubled; when Q is reversed, R is again doubled.
Find: P: Q: R vectors.
Solution:
Using the formula of vectors, we get:
R² = P² + Q² +2|P||Q|Cosθ ---------- (1)
Q is doubled, R is doubled, so:
(2R)² = P² + (2Q)² +2|P||2Q|Cosθ
Solving, we get: 4R² = P² + 4Q² +4|P||Q|Cosθ ---------- (2)
Q is reversed, R is again doubled, so:
(2R)² = P² + (-Q)² +2|P||-Q|Cos(180-θ)
Solving, we get:
4R² = P² + Q² -2|P||Q|Cosθ ---------- (3)
- Reversing a vector means changing its direction by 180.
- |-Q| = |Q|
- Cos(180-θ) = -Cosθ
Subtracting (3) from (2), we get:
0 = 3Q² + 6|P||Q|Cosθ
Assuming Q as non-zero vector,
-3|Q| = 6|P|Cosθ
|Q| = -2|P|Cosθ ---------- (4)
Squaring both sides, Q² = 4P².(Cosθ)² ---------- (5)
Subtracting (1) from (2), we get:
3R² = 3Q² + 2|P||Q|Cosθ ----------(6)
Substituting the value obtained from (5) and (4) in (6):
3R² = 3[4P²(Cosθ)²] + 2|P|[-2|P|Cosθ]Cosθ
3R² = 12P²(Cosθ)² - 4P²(Cosθ)²
3R² = 8P²(Cosθ)² ---------- (7)
Substituting the values of R² and Q², obtained from (4) & (5) in (1):
[8P²(Cosθ)²]3 = P² + [4P²(Cosθ)²] + 2|P|[-2|P|Cosθ]Cosθ
16P²(Cosθ)² = P² + 4P²(Cosθ)² – 4P²(Cosθ)²
16P²(Cosθ)² = P²
Assuming P as non-zero vector,
(Cosθ)^2 = 1/16 ---------- (8)
We know from (5), Q² = 4P²(Cosθ)²
From (7), R² = [8P²(Cosθ)²]/3
Put value of (Cosθ)² obtained from (8), we get:
Q² = 1/4*(P²)
R² = 1/6*(P²)
So P² : Q² : R² = P² : P²/4 : P²/6 = 1: 1/4 : 1/6 = 1: 0.25 : 0.16
Ratio of P:Q: R = 1 : 0.05 : 0.04