Angle Between Two Line
Angle Between Two Line
The angle between two lines in a plane is defined to be
<-- 0, if the lines are parallel;
<-- the smaller angle having as sides the half-lines starting from the intersection
point of the lines and lying on those two lines, if the lines are not parallel.
If denotes the angle between two lines, it always satisfies the inequalities
This equation clicks in the case that m1m2=-1 , when the lines are perpendicular
and equals to 2 .
Also, if one of the lines is parallel to y -axis, it has no slope; then the angle must
be deduced using the slope of the other line.
If one of the slopes is 0 , the angle between the two lines is just the angle between
one of the lines and the x -axis. Assume the other line has slope m , then formula
(2) above becomes
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If, on the other hand, one of the slopes is infinite, meaning that the line is parallel
to the y -axis, then the angle between two lines is the same as the angle between
one line (with slope m ) and the y -axis, which is
The above formula is consistent with formula (2) in the sense that if we let one of
m1 or m2 approach , we get formula (4).
Remark. If both slopes are positive, then formula (2) above is really just a
disguised form of the subtraction formula for tangent.
In the diagram above, we see that the angle between the two lines is the algebraic
difference of the two angles made between each of the lines and the x -axis.
In the Euclidean space, the angle between two lines is most comfortably defined
by using the direction vectors u and v of the lines:
cos= uvuv

Also this angle satisfies (1). The angle given by the cosine can be interpreted to be
formed after translating the one line in the space, without to alter its direction, to
such a location that it intersects the other line -- then both lines are in the same
plane, and one may think that the angle is defined as in the beginning of this
entry.
Remark. The angle between two curves which intersect each other in a point P
means the angle between the tangent lines of the curves in P ; such an angle may
always be chosen acute or right.
For example, the exponential curves y=ax and y=bx intersect each other in the
point (01) under the angle with tan=lna-lnb1+lnalnb .
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Let slope of one line is zero and slope of other line is `a', then we get the formula
which is given below:
=> Tan = |a|;
Suppose one of the slope is infinite and that line is parallel to the y - axis, then the
angles between both the lines is same as the angle between one line (which has
slope `a') and y - axis.
So the formula for this statement is given below:
=> Tan = |1 / m|,
We get this formula when one of the slope a1 a2 approaches to .
We find the Angle between two lines by using vector product method by using
formula which is given below:
Cos =


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