Find the angles between lines $y=3x-4$ and $y = 5-2x$.
First I represented the lines into their respective vector equation and then I took the dot product. And solved for the angle. But do we take the largest angle or does it matter?
In this case, I got $\theta = 135 $ but the answer gave $\theta = 45$. Am I wrong?
Should I take red angle or blue angle?
On
There are three different but related concepts: angle between two lines, angle between two vectors and angle from one vector to another vector. Taking the dot product of two vectors and dividing by the product of their norms will give the angle between the vectors but not not the directed angle, i.e. the angle from one vector to the other, Taking vectors along the two lines and finding the angle between them is arbitrary because a line has no inherent direction-we could have taken one or both vectors in the opposite direction, which explains why you cannot meaningly distinguish beween an angle and its suplementary angle when speaking of the angle between two lines.
The issue is purely a matter of semantics, so it's not really possible for us to give a conclusive answer. It ultimately depends on what the author of the question desired.
That said, usually with such questions, my experience makes me assume it's the smaller of the two angles unless it's explicitly noted otherwise (i.e. in your case, $45^\circ$).
It would've been nice if the author was more specific, though. But for all intents and purposes in my eyes, you're right nonetheless. You just didn't give the answer the author wanted, but it is still the angle between two of the lines nonetheless.