Answer to Question #70408 in Geometry for sajid

Question

Q.Prove that If the tangent vector of the parameterized curve is constant, the image of the curve is (part of) a straight line.

Answer 1

The equation of the parameterized curve is {x=x(t), y=y(t), t in (a,b)}.The tangent vector is ( x’(t), y’(t)), t in (a,b), if it is constant then x’(t)=C1, y’(t)=C2, where C1 and C2 are constants. Solving these differential equations, we obtain x(t)=C1t+C3, y(t)=C2t+ C4, t in (a,b), where C3 and C4 are constants. It is mean that the curve is a straight line, because if C1≠0 then y(x)=C2 (x- C3)/C1 + C4 =(C2 /C1) x- C3/C1 +C4 is a straight line, if C2≠0 then x(y)=C1 (y- C4)/C2 + C3 =(C1 /C2) y- C4/C2 +C3 is a straight line,