# 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

- becomes
## 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 lines

If 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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