if the line L1, passes through the first pair of points, L2 through the second pair, find the angle from L1 to L2.
got this ffrom the book analytic geometry by love and rainville and it give me 45 degrees as an answer. how is that?
You need to remember the equation that helps you to find the angle between the given lines `l_1` and `l_2` , such that:
`tan alpha = (m_1 - m_2)/(1 + m_1*m_2)`
`m_1` represents the slope of the line ` l_1`
`m_2` represents the slope of the line ` l_2`
Since the line l_1 passes through the points `(1,9)` and `(2,6)` yields:
`m_1 = (6 - 9)/(2 - 1) => m_1 = -3`
Since the line `l_2` passes through the points `(3,3)` and `(-1,5)` yields:
`m_2 = (5 - 3)/(-1-3) => m_2 = 2/(-4) => m_2 = -1/2`
You may substitute -`3` for `m_1` and -`1/2` for `m_2` in equation of tangent `alpha` , such that:
`tan alpha = (-3 + 1/2)/(1 + 3/2) => tan alpha = (-5/2)/(5/2) = -1`
You need to remember that the tangent of an angle is negative if the angle is obtuse, hence `alpha = pi - pi/4 = (3pi)/4` or `alpha = 180^o - 45^o = 135^o` .
Hence, evaluating the angle between the lines l_1 and l_2 yields `alpha = (3pi)/4` .
The formula for finding the angle from first line to the second line is given by:
tan θ = (m1-m2)/(1+m1m2)
provided that neither L1 or L2 is // to y-axis
where: m1 is the slope of firt line & m2 is the slope of the second line
For L1: m1 = (y2-y1)/(x2-x1)
m1 = (6-9)/(2-1) = -3
For L2: m2 = (y2-y1)/(x2-x1)
m2 = (5-3)/(-1-3) = -1/2
tan θ = [-3-(-1/2)]/[1+(-3)(-1/2)]
tan θ = -1
θ = Arctan (-1)
θ = -45°
The negative result indicates that the acute angle from L1 to L2 is measured counterclockwise direction. Recognising that there are distinct angles formed between two intersecting lines, one acute and obtuse except when the lines are perpendicular to each other, the result of Arctan using calculator is always acute. The obtuse angle my be obtained by 180-θ but opposite direction.
The acute angle between L1 & L2 is 45° and
The obtuse angle between L1 & L2 is 135°
The acute angle from L1 to L2 is 45° counterclockwise and
The obtuse angle from L1 to L2 is 135° clockwise