Show that both the given lines are parellel (3-D)

Question

Guys I am trying to solve the problem below which I found in a book:

Show that the lines $\frac{x-1}{2}=\frac{2-y}{1}=\frac{5-z}{1}$ and $\frac{4-x}{4}=\frac{3+y}{2}=\frac{5+z}{2}$ are parallel.

I am trying to solve it by finding direction vectors for the lines and then calculating their dot product.It should be $1$ ideally for them too be parallel.

I know I am missing something in my method of solving it.Please help.

Answer 1

The dot product of two vectors is 0 if and only if they are perpendicular, not parallel. The direction vectors of the two lines are <2, -1, -1> and <-4, 2, 2>. It is easily seen that they are scalar multiples of each other and so the lines must be parallel.(Or you can show that cos(a) = 1 as in the answer given).

Answer 2

$$\cos(a) = \frac{2\cdot4+1\cdot2+1\cdot2}{\sqrt{6}\sqrt{24}} = 1$$

Thus, vectors are parallel.