Two vectors of magnitude 2p and p if first one is doubled without changing the intermediate result,then the intermediate became 3 times. Find the intermediate angle.

To solve the problem of finding the intermediate angle between two vectors of magnitudes 2p and p, we need to delve into some vector mathematics. The scenario describes a situation where the first vector is doubled, and as a result, the intermediate becomes three times its original value. Let's break this down step by step.

Understanding the Vectors

We have two vectors, which we can denote as:

• Vector A: Magnitude = 2p
• Vector B: Magnitude = p

Let’s denote the angle between these two vectors as θ. The resultant vector R when these two vectors are added can be calculated using the law of cosines:

Law of Cosines for Vectors

The magnitude of the resultant vector R is given by:

R = √(A² + B² + 2AB cos(θ))

Substituting the magnitudes of our vectors:

R = √((2p)² + p² + 2(2p)(p) cos(θ))

This simplifies to:

R = √(4p² + p² + 4p² cos(θ))

or

R = √(5p² + 4p² cos(θ))

Doubling the First Vector

Now, if we double the first vector, its new magnitude becomes 4p. The new resultant vector R' can be calculated similarly:

R' = √((4p)² + p² + 2(4p)(p) cos(θ))

This simplifies to:

R' = √(16p² + p² + 8p² cos(θ))

or

R' = √(17p² + 8p² cos(θ))

Setting Up the Equation

According to the problem, when the first vector is doubled, the intermediate result becomes three times the original resultant. Therefore, we can set up the equation:

R' = 3R

Substituting our expressions for R and R', we have:

√(17p² + 8p² cos(θ)) = 3√(5p² + 4p² cos(θ))

Squaring Both Sides

To eliminate the square roots, we square both sides:

17p² + 8p² cos(θ) = 9(5p² + 4p² cos(θ))

Expanding the right side gives:

17p² + 8p² cos(θ) = 45p² + 36p² cos(θ)

Rearranging the Equation

Now, let’s rearrange the equation to isolate terms involving cos(θ):

17p² - 45p² = 36p² cos(θ) - 8p² cos(θ)

This simplifies to:

-28p² = 28p² cos(θ)

Dividing both sides by 28p² (assuming p is not zero) gives:

cos(θ) = -1

Finding the Angle

The cosine of an angle equals -1 at an angle of 180 degrees. Thus, the intermediate angle θ between the two vectors is:

θ = 180°

In summary, when the first vector is doubled, the angle between the two vectors that results in the intermediate becoming three times its original value is 180 degrees. This means the two vectors are pointing in exactly opposite directions.