Question

Resultant of $\vec{P}$ and $\vec{Q}$ is $\vec{R}$. If $\vec{Q}$ is doubled, $\vec{R}$ is doubled but when $\vec{Q}$ is reversed(-Q), $\vec{R}$ is again doubled. Find P:Q:R.​

Answer 1

Solution :

Let, θ be the angle between $\vec{P}$ and $\vec{Q}$

Given that, $\vec{R}$ is resultant

Then, R² = P² + Q² + 2PQcosθ ...(i)

When $\vec{Q}$ is doubled, $\vec{R}$ gets doubled. Then,

(2R)² = P² + (2Q)² + 2P (2Q) cosθ

⇒ 4R² = P² + 4Q² + 4PQcosθ ...(ii)

When $\vec{Q}$ is reversed, $\vec{R}$ gets doubled. Then

(2R)² = P² + Q² + 2PQ cos(π - θ)

⇒ 4R² = P² + Q² - 2PQcosθ ...(iii)

Adding (i) and (iii), we get

R² + 4R² = P² + Q² + 2PQcosθ + P² + Q² - 2PQcosθ

⇒ 5R² = 2P² + 2Q² ...(iv)

Adding (ii) and {(iii) × 2}, we get

4R² + 8R² = P² + 4Q² + 4PQcosθ + 2P² + 2Q² - 4PQcosθ

⇒ 12R² = 3P² + 6Q²

⇒ 4R² = P² + 2Q² ...(v)

Now, (iv) - (v) gives

R² = P²

⇒ R²/P² = 1/1

⇒ R/P = 1/1

⇒ R : P = 1 : 1

⇒ R : P = √2 : √2

Putting R² = P² in (v), we get

4R² = R² + 2Q²

⇒ 3R² = 2Q²

⇒ R²/Q² = 2/3

⇒ R/Q = √2/√3

⇒ R : Q = √2 : √3

Therefore, P : Q : R = √2 : √3 : √2

Answer 2

Solution:

Suppose theta be 'A'

Let A be an angle between vector P and vector Q.

Given,

vector R is resultant.

R^2 = P^2 + Q^2 + 2PQ cosA _____(1)

When vector Q is doubled, vector R is also doubled.

(2R)^2 = P^2 + (2Q)^2 + 2P × (2Q)^2 × cosA

=> 4R^2 = P^2 + 4Q^2 + 4PQ cosA ____(2)

Now,

When vector Q is reversed, vector R is also doubled.

(2R)^2 = P^2 + Q^2 - 2PQ cos(π - A)

=> 4R^2 = P^2 + Q^2 - 2PQ cosA ______(3)

By adding equation (1) and (3), we get

R^2 + 4R^2 = P^2 + Q^2 + 2PQ cosA + P^2 + Q^2 - 2PQ cosA

Here, ( + 2PQ cosA) and ( - 2PQ cosA) is canceled.

R^2 + 4R^2 = P^2 + Q^2 + P^2 + Q^2

We get,

=> 5R^2 = 2P^2 + 2Q^2 _______________(4)

Now,

equation (4) and (5) gives,

R^2 = P^2

=> R^2 / P^2 = 1/1

=> R/P = 1/1

=> R:P = 1:1

So, R:P = √2:√2

Putting R^2 = P^2 in equation (5), we get

4R^2 = R^2 + 2Q^2

=> 4R^2 - R^2 = 2Q^2

=> 3R^2 = 2Q^2

=> R^2 / Q^2 = 2/3

=> R / Q = √2/√3

=> R : Q = √2:√3

Therefore, P : Q : R = √2 : √3 : √2

Thus, the ratio of P : Q : R = √2 : √3 : √2.