Q.Find the values of $c$ so that for all real $x$ the vectors $xc\hat i-6\hat j+3\hat k$ and $x\hat i+2\hat j+2cx\hat k$ make an obtuse angle.

Let $\bar p=xc\hat i-6\hat j+3\hat k,\ \bar q=x\hat i+2\hat j+2cx\hat k$. Their dot product is

For the angle between $\bar p,\bar q$ to be obtuse for all real $x$, we need $\bar p\cdot\bar q<0$ for every

$x$, i.e. $cx^2+6cx-12<0$ for all $x$.

A quadratic $f(x)=cx^2+6cx-12$ is negative for all $x$ exactly when its leading coefficient is negative

($c<0$, so the parabola opens downward) and its discriminant is negative (so it never touches or crosses

zero):

Combined with $c<0$ (already implied), the condition is $-\dfrac43<c<0$.

Form the dot product as a quadratic in x, then require it to be negative for all real x, which needs a negative leading coefficient and negative discriminant; solve the resulting compound inequality for c.

Forgetting to also require the leading coefficient c<0 (only checking the discriminant condition), which can let in values of c where the parabola opens upward and eventually becomes positive.