How do you find the acute angle between two intersecting lines whos equations are given in vector form?

To figure out the acute angle between two intersecting lines given in vector form, first, let's focus on which angle we're after. When two lines cross, they form two pairs of equal and opposite angles, making four in total. We're interested in finding the smaller one of these. The trick here is to use the dot product formula, but in a rearranged form. You don't need the whole line equations for this, just the direction vectors – these are the bits alongside the scalar multipliers in the equations. Calculate the dot product of these direction vectors and also work out their magnitudes. Once you have these, plug them into your rearranged dot product formula. To get the angle from here, apply the inverse cosine (arccos) function to this value, and voilà, you've got the acute angle you were looking for!