Set \theta = 90^\circ in the angle formula. Since \cos 90^\circ = 0, the whole dot product collapses to zero. That gives the cleanest, most useful test in all of linear algebra:

Two non-zero vectors are orthogonal — a fancy word for perpendicular — exactly when their dot product is zero. No angles, no cosines, no square roots: just multiply, add, and check for 0. (The zero vector is taken to be orthogonal to everything.)

Hunt for the right angle

Keep \vec{u} fixed and rotate \vec{v}. Watch the dot-product readout: the little square pops up at the origin exactly when it hits 0. That's the only moment the arrows are truly perpendicular.

Why we love perpendicular

Orthogonal directions don't interfere with each other — moving along one changes nothing about your position along the other. That independence is why x and y axes are drawn at right angles, why \hat{\imath}\cdot\hat{\jmath} = 0, and why the cleanest bases are built from mutually orthogonal unit vectors. A quick check: any vector \begin{bmatrix} a \\ b \end{bmatrix} is orthogonal to \begin{bmatrix} -b \\ a \end{bmatrix} — swap and negate to turn a right angle.