Resultant of 2 vectors P & Q is inclined at 45 to either of them. What is the magnitude of resultant vector?

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To find the magnitude of the resultant vector when two vectors P and Q are inclined at 45 degrees to either of them, you can use the Pythagorean theorem and trigonometric identities.

Let's assume the magnitude of vector P is |P| and the magnitude of vector Q is |Q|. Since the resultant vector is inclined at 45 degrees to either of them, we can create a right triangle with vector P and vector Q as the perpendicular and base, respectively.

According to the Pythagorean theorem, the magnitude of the resultant vector (R) can be found using the formula:

R^2 = |P|^2 + |Q|^2

Since the vectors are inclined at 45 degrees, the magnitudes can be related to the components using trigonometric identities:

|P| = P * cos(45) = P * sin(45) (since cos(45) = sin(45) = 1/sqrt(2))

|Q| = Q * cos(45) = Q * sin(45) (since cos(45) = sin(45) = 1/sqrt(2))

Substituting these values back into the Pythagorean theorem formula:

R^2 = (P * sin(45))^2 + (Q * sin(45))^2

R^2 = (P^2 * sin^2(45)) + (Q^2 * sin^2(45))

R^2 = (1/2) * (P^2 + Q^2)

Taking the square root of both sides, we get:

R = sqrt((1/2) * (P^2 + Q^2))

Therefore, the magnitude of the resultant vector is given by:

|R| = sqrt((1/2) * (P^2 + Q^2))