Problem Statement:
The resultant of vector p and q is r. If the magnitude of vector q becomes double, then the resultant becomes perpendicular to vector p. Find the magnitude of r.

Solution:
Let the magnitude of vector p be 'a' and the magnitude of vector q be 'b'.
Given that the resultant of vector p and q is r. Therefore, we can write:
r = √(a^2 + b^2) ------- Equation 1

When the magnitude of vector q becomes double, let the new magnitude be '2b'. At this point, the resultant becomes perpendicular to vector p. Therefore, the dot product of vector p and the new vector q (2q) is equal to zero.
p * 2q = 0
p * (2b) = 0
2p * b * cos(θ) = 0 (where θ is the angle between p and q)
p * b * cos(θ) = 0
p * q = 0

From the above equation, we can say that vector p and vector q are perpendicular to each other. Therefore, we can write:
p * q = a * b * cos(90) = 0
a * b = 0

As a and b cannot be zero, we can conclude that the angle between vector p and q is 90 degrees.

Now, let's find the magnitude of r when the magnitude of vector q becomes double.
Using the Pythagorean theorem, we can write:
r^2 = a^2 + (2b)^2 = a^2 + 4b^2
r = √(a^2 + 4b^2) ------- Equation 2

Substituting Equation 1 in Equation 2, we get:
r = √(r^2 - b^2) * 2
r^2 = 4r^2 - 4b^2
3r^2 = 4b^2
r^2 = (4/3) * b^2
r = √((4/3) * b^2)

Therefore, the magnitude of r is √((4/3) * b^2).