What vector represents the line? One way to find it is to use the old-fashioned slope intercept form. y = (-a/b)x - c/b
This gives such a vector (1,-a/b). The vector product should be scalar zero.
Good Day,
I've been asked to show that the nonzero vector n = (a, b) is perpendicular to the line ax+by+c=0. I know how to prove this by using gradients but I'd like to show it using vectors.
I've seen a similar post regarding this but I wasn't able to understand it 'cos the methods weren't clearly shown...
Assistance is greatly appreciated.
What vector represents the line? One way to find it is to use the old-fashioned slope intercept form. y = (-a/b)x - c/b
This gives such a vector (1,-a/b). The vector product should be scalar zero.
Hi. Thanks for the reply.
However, there's a way to solve this by considering two points on the line, P1 = (x1, y1) and P2 = (x2, y2).
The vector P1P2 is found as such (x2-x1, y2-y1).
After this, one finds the dot/ scalar product of the vectors n and P1P2. The result is ax2 - ax1 + by2 - by1.
What I don't get is how the result of the dot product is 0, therefore proving that the vector n is perpendicular to the line ax + by + c = 0.
Please do provide detailed steps to help me understand.
You have vector (a,b) you need to find vector (k,t) perpendicular to vector (a,b).
(a,b)*(k,t)=0
ak+bt=0
a/b=-t/k
Hence: t/k=(-a/b) ==> t=-ma, k=mb where m is scalar...
choose m=1:
you will get that: vector (k,t)=(b,-a)
(b,-a)=(-b,a)=(1,-a/b)