Animated Solution for Physics - Kinematics: If $\mathbf{A}$ and $\mathbf{B}$ are two vectors satisfying the relation $\mathbf{A}\cdot \mathbf{B}=|\mathbf{A}\times \mathbf{B}|$. Then, the value of $|\mathbf{A}-\mathbf{B}|$ will be

Vectors are the language of physics, allowing us to describe quantities that have both magnitude and direction. Two of the most fundamental operations we can perform on vectors are the dot product and the cross product.

The dot product, defined as $\mathbf{A}\cdot \mathbf{B}=AB\cos \theta$, tells us how much of one vector goes in the direction of the other. It's a measure of parallelism. On the other hand, the magnitude of the cross product, $|\mathbf{A}\times \mathbf{B}|=AB\sin \theta$, tells us how much of one vector is perpendicular to the other.

In this problem, we are given a beautiful symmetric condition: the dot product equals the magnitude of the cross product.

Substituting our definitions into this equation, we get:

Assuming our vectors are non-zero, we can divide both sides by $AB$. This leaves us with a simple trigonometric equation:

Dividing by $\cos \theta$, we find that $\tan \theta =1$. The only angle between $0^{\circ }$ and $180^{\circ }$ that satisfies this is $\theta =45^{\circ }$. This means our vectors are perfectly balanced between being parallel and perpendicular!

Now that we know the angle between the vectors, we need to find the magnitude of their difference, $|\mathbf{A}-\mathbf{B}|$. Geometrically, if we place $\mathbf{A}$ and $\mathbf{B}$ tail-to-tail, $\mathbf{A}-\mathbf{B}$ is the vector pointing from the tip of $\mathbf{B}$ to the tip of $\mathbf{A}$.

Alternatively, we can think of it as adding $\mathbf{A}$ and $-\mathbf{B}$. Using the law of cosines for vector addition, the magnitude of the difference is given by:

We simply substitute our known angle of $45^{\circ }$ into the formula:

Since $\cos (45^{\circ })=\frac{1}{\sqrt{2}}$, we have:

Simplifying the fraction $\frac{2}{\sqrt{2}}$ gives us $\sqrt{2}$. Thus, our final expression is:

This elegant result perfectly captures the geometric relationship between the two vectors.