Animated Solution for Physics - Kinematics: Two vectors $\mathbf{P}$ and $\mathbf{Q}$ have equal magnitudes. If the magnitude of $\mathbf{P}+\mathbf{Q}$ is $n$ times the magnitude of $\mathbf{P}-\mathbf{Q}$, then angle between $\mathbf{P}$ and $\mathbf{Q}$ is

Imagine you are looking at two vectors, $\mathbf{P}$ and $\mathbf{Q}$, originating from the exact same point. The problem tells us a very specific and beautiful constraint: their magnitudes are perfectly equal. Let's call this common magnitude $P$.

Now, when we add these two vectors, $\mathbf{P}+\mathbf{Q}$, we are geometrically finding the main diagonal of the parallelogram formed by them. Conversely, when we subtract them, $\mathbf{P}-\mathbf{Q}$, we are finding the length of the other diagonal. The question states that the length of the main diagonal is $n$ times the length of the other diagonal.

Let's translate this geometric reality into pure algebra. The magnitude of the sum and difference of two vectors separated by an angle $\theta$ are given by standard formulas:

According to the problem, $|\mathbf{P}+\mathbf{Q}|=n|\mathbf{P}-\mathbf{Q}|$. Substituting our formulas into this condition gives us:

Square roots can be visually intimidating and algebraically clunky. Let's square both sides to shatter them. Don't forget to square the $n$ as well!

Here is where the magic happens. We know that the magnitudes are equal, so we can substitute $Q$ with $P$ everywhere in the equation.

Notice how every single term now contains a $2P^2$. We can factor this out on both sides:

Since the vectors are non-zero, $2P^2$ is not zero, and we can safely cancel it out entirely. The equation collapses into something incredibly elegant:

Our goal is to isolate $\theta$. Let's expand the right side and group all the $\cos \theta$ terms together.

Moving the cosine terms to the left and the constants to the right:

Finally, divide by $(1+n^2)$ to isolate $\cos \theta$:

Taking the inverse cosine gives us our final angle:

A Geometric Thought Experiment: What if $n=1$? This would mean the diagonals of our parallelogram are perfectly equal. Our formula would give $\cos \theta =0$, meaning $\theta =90^{\circ }$. A parallelogram with equal adjacent sides (a rhombus) that has equal diagonals is strictly a square. Physics and geometry are always in perfect harmony!