Can magnitude of a vector can be negative?

Question

In the answer, at last why we have taken (2-t)=±1 ? I think magnitude can only be positive so (2-t)=1 only .

(Feel free to edit if any)

Answer 1

You are correct that the magnitude of a vector must ALWAYS be positive. However, in the problem they are solving for dot product divided by magnitude and dot product can be negative. Also note the absolute value in the numerator, that's where the plus or minus comes from.

Answer 2

Magnitude is a many-one function, you give a number(either positive or negative), it returns the size of the number(absolute value). The reason $2-t = \pm 1$ is you are trying to find the x for which f(x) = 1, which means x can be $\pm1$, in this case x is $2-t$.

Hope this helps.