How to change the constant C in the general equation of the straight line Ax+By+C=0 when the straight line is shifted along the vector?

Question

Mathematics

There is a straight line given by the general equation of the straight line Ax+By+C=0, there is a vector V. It is
necessary to shift the straight line along this vector V (change the origin of the coordinate system of the straight line?), i.e. A and B does not change, how to calculate C?

Answer 1

If V = (vx, vy), then points (x+vx, y+vy) will satisfy the equation of the new line instead of (x, y):

Ax+By+C=0
A(x+vx)+B(y+vy)+C'=0
C' = C-Avx-Bvy

Answer 2

U=(A,B) is the perpendicular vector of the line.
Then dC = Pr _U V·|U| = U·V
Just the scalar product.
Examination. x + y = 0, U=(1,1), V = (1,2)
dC = 1 1 + 1 2 = 3, right?