I have two lines
a) for what values of k are the lines perpendicular?
I got this by using the dot product of [4,-5].[7,k]=0 and then solving for k.
b) for what value of k are the lines paralell?
I'm not sure how to do this. I know the the normal vector is n=[A,B] and the direction vector is m=[B,-A].
so I took the dot product of [5,4].[7,k] =1
I made the dot product equal to 1 since cos 0 is 1.
But much simpler is that if two vectors are parallel, one is a multiple of the other. You must have [5,4]= m[7, k] so you have two equations, 5= 7m and 4= mk to solve for the two unknowns. Since you really only want k, write m= 5/7, from the first equation, and replace m by 5/7 in the second equation.