and at the origin.Is this last statement is correct? If yes, please give an example that two curves are cut at right angles and the product of the slopes m1*m2 is not equal to -1.
The angle between the curves C1 and C2 at a point of intersection P is
defined to be the angle between the tangent lines to C1 and C2 at P (if these
tangent lines exist) Let us represent the two curves C1 and C2 by the Cartesian
equation y = f(x) and y = g(x) respectively. Let them intersect at P (x1,y1) .
If ψ1 and ψ2 are the angles made by the tangents PT1 and PT2 to
C1 and C2 at P, with the positive direction of the x – axis, then m1 = tan ψ 1 and
m2 = tan ψ2 are the slopes of PT1 and PT2 respectively.
Let ψ be the angle between PT1
and PT2. Then ψ = ψ2 – ψ1 and
tan ψ = tan (ψ2 – ψ1)
=
tan ψ2 – tan ψ1
1 + tan ψ1 tan ψ2
=
m2 – m1
1 + m1m2
where 0 ≤ ψ < π
We observe that if their slopes are equal namely m1 = m2 then the two
curves touch each other. If the product m1 m2 = – 1 then these curves are said to
cut at right angles or orthogonally. We caution that if they cut at right angles
then m1 m2 need not be –1.
We caution that if they cut at right angles
then m1 m2 need not be –1.
Is this last statement is correct? If yes, please give an example that two curves are cut at right angles and the product of the slopes m1*m2 is not equal to -1.
Is this two curves and at the origin are at right angle? I don't know to prove that these two curves are at right angles. Please help me to prove that too..
Another one question... Is there any difference between perpendicular and orthogonal? If yes let me what is the difference is?
I had come across a statement that if m1*m2=-1, then two curves are at right angles. The converse need not be true. i.e, if two cuves are at right angles, then m1*m2 may or may not be -1.
Thats why i need curves that are at right angles at a point and slopes m1*m2 not equals -1. You have given two curves y=x^3 and y=x^(1/3) at origin. I checked m1*m2 not equals -1. But these cuves are at right angles at origin? Please explain it.....
Then, when i gone for perpendicular and orthogonal in the web i got that two terms are the same... Then for what purpose we are using the two terms for single concept "RIGHT ANGLES"?
Kind of for helping....
once again ... what do you know about finding the slope of a curve at a specific point?
they are synonyms ... the term orthogonal is usually used more often with vectors.Then, when i gone for perpendicular and orthogonal in the web i got that two terms are the same... Then for what purpose we are using the two terms for single concept "RIGHT ANGLES"?