Cartesian Form of a line passing through a given point
Equation of a line passing through a given point P(x1, y1, z1) and parallel to a given vector b having direction ratios <a, b, c> is
If <l, m, n> are the direction cosines then its equation is given by
Reduction of Cartesian form to the Vector form
If is the equation of line then equate this to a
constant, say λ to get the vector form.
So, = λ
Hence the vector equation of line passing through a point with position vector a and parallel to b is r=a+λb
Reduction of Vector form to the Cartesian Form
= λ, which is the Cartesian form
Cartesian equation of a line passing through two points
If P(x1, y1, z1) and Q(x2, y2, z2) are two given points then Cartesian equation of line is
Reduction of Cartesian form to Vector form
As in the previous case, equate the Cartesian form of the line to λ, so that it
Reduction of Vector form to Cartesian form
Hence from the above three equations, we can write
= λ, which is the Cartesian form.
Angle between two linesIf are the vector equations of two lines then angle between them is given by
Equations of two lines then angle between them is given by
Condition for parallelism and perpendicularity• If the lines are parallel then
• If the lines are perpendicular then a1a2+b1b2+c1c2=0
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