I have two lines
r: [x,y]=[3,2]+t[4,-5]
q: [x,y]=[1,1]+s[7,k]
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
35+4k=1
k=-(34/4)
I made the dot product equal to 1 since cos 0 is 1.
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The dot product of two vectors, a and b, is . Of course, if the lines are parallel, . If you really want to use the dot product, you would want it equal to , not 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.