What does a negative dot product tell you about two vectors?

The right-hand side has two lengths and a cosine. Lengths are never negative, so after dividing by $\lVert u\rVert\lVert v\rVert$, assuming neither vector is zero, the sign of $u\cdot v$ is exactly the sign of $\cos\theta$. The cosine is positive for angles below $90^\circ$, zero at $90^\circ$, and negative for angles between $90^\circ$ and $180^\circ$. A negative dot product therefore places the angle in that last interval. It does not say the angle is exactly $180^\circ$: that extreme is reached only when the vectors are exact opposites and the dot product equals $-\lVert u\rVert\lVert v\rVert$.

Take $u=(1,0)$ and $v=(-1,1)$. The coordinatewise products are $1\times(-1)=-1$ and $0\times1=0$, so $u\cdot v=-1$. The lengths are $\lVert u\rVert=\sqrt{1^2+0^2}=1$ and $\lVert v\rVert=\sqrt{(-1)^2+1^2}=\sqrt2\approx1.414$. Dividing gives $\cos\theta=-1/1.414\approx-0.707$, and $\theta\approx135^\circ$. The vectors are not pointing in exactly opposite directions, but the angle is $45^\circ$ past a right angle, which is what the negative sign reports.

You can also read the calculation as a tally of agreement and disagreement coordinate by coordinate. Matching signs contribute positive products; opposing signs contribute negative products. The total is net agreement, so a negative total means the opposing contributions outweigh the aligned ones. That is why a single strongly opposing coordinate can make the whole dot product negative even when another coordinate agrees, as with $u=(2,1)$ and $v=(-1,1)$, whose products are $-2$ and $1$ and whose sum is $-1$.

The common misreading is to treat a negative value as "no similarity" or as a failed comparison. Zero is the value that means perpendicular, and perpendicular is the usual arithmetic picture of two unrelated directions. Negative is stronger and more specific: the vectors are on opposite sides of perpendicular. Nor does a negative sum require every coordinate pair to oppose. One coordinate can agree and another can disagree strongly enough to make the total negative, so the sign describes the net direction of the pair, not a uniform opposition in every coordinate.

The sign survives scaling; the magnitude does not. Keep $u=(1,0)$ and replace $v=(-1,1)$ by $v'=(-2,2)$. Then $u\cdot v'=1\times(-2)+0\times2=-2$, twice as negative as before. But $\lVert v'\rVert=\sqrt{(-2)^2+2^2}=\sqrt8\approx2.828$, so $\cos\theta=-2/2.828\approx-0.707$, the same cosine and the same $135^\circ$ angle. The extra negative size came from the longer second vector, not from a more opposite direction. This is why the sign is safe to read as a side of perpendicular while the raw number is not safe to compare across vectors of different lengths.

The reading fails when either vector is zero, because the angle is undefined and the formula divides by a zero length. It also cannot occur if every coordinate in both vectors is non-negative: every product is then non-negative and the sum is non-negative, so no obtuse angle is available. In many dimensions, two unrelated vectors tend to have a dot product near zero rather than clearly negative; a negative value is evidence of opposition, not merely evidence of dissimilarity.

If both vectors have length one, the division by lengths does nothing and the dot product is the cosine itself. A negative dot product then has magnitude at most one, and its value is the cosine of the obtuse angle. For general vectors, recover the angle with $\theta=\arccos\left(\frac{u\cdot v}{\lVert u\rVert\lVert v\rVert}\right)$. The sign of the inside decides the answer: negative inside means obtuse, zero means right, positive means acute.