Animated Solution for Physics - Kinematics: If $\mathbf{P}\times \mathbf{Q}=\mathbf{Q}\times \mathbf{P}$, the angle between $\mathbf{P}$ and $\mathbf{Q}$ is $\theta$ ($0^{\circ }<\theta <360^{\circ }$). The value of $\theta$ will be ……………

Imagine you are playing with two vectors, $\mathbf{P}$ and $\mathbf{Q}$. You decide to take their cross product, $\mathbf{P}\times \mathbf{Q}$. Then, just for fun, you reverse the order and calculate $\mathbf{Q}\times \mathbf{P}$. To your surprise, the results are exactly the same!

This is a highly unusual situation. Why? Because the cross product is inherently anti-commutative. Let's dive into the mathematics to uncover what this implies about the angle between our two vectors.

By definition, the cross product follows the right-hand rule. If you curl your fingers from $\mathbf{P}$ to $\mathbf{Q}$, your thumb points in the direction of $\mathbf{P}\times \mathbf{Q}$. If you curl them from $\mathbf{Q}$ to $\mathbf{P}$, your thumb points in the exact opposite direction. Mathematically, this is written as:

Now, let's look at the condition given in our problem:

We can substitute our anti-commutative property directly into the given equation. Replacing the right side, we get:

This looks like a simple algebraic equation of the form $x=-x$. Let's bring all terms to one side by adding $(\mathbf{P}\times \mathbf{Q})$ to both sides:

Dividing by 2, we arrive at a profound conclusion:

The cross product of two vectors is zero. What does this mean physically? Let's expand the magnitude of the cross product:

Assuming $\mathbf{P}$ and $\mathbf{Q}$ are non-zero vectors (otherwise the angle $\theta$ wouldn't be well-defined), the only way this product can be zero is if the sine of the angle between them is zero:

The sine function is zero at integer multiples of $180^{\circ }$ (or $\pi$ radians). So, $\theta$ could be $0^{\circ }$, $180^{\circ }$, $360^{\circ }$, etc.

However, the problem provides a strict domain for the angle: $0^{\circ }<\theta <360^{\circ }$.

Because the angle must be strictly greater than $0^{\circ }$ and strictly less than $360^{\circ }$, the only valid solution is:

This means the vectors $\mathbf{P}$ and $\mathbf{Q}$ are anti-parallel; they point in exactly opposite directions. It's a beautiful example of how a simple algebraic property of vectors can dictate their geometric orientation in space!