# 3D Geometry - 2

__ Illustration__: Find angle b/w lines whose direction cosines are & .

Ans: Let be angle

cos = l

_{1}l

_{2}+ m

_{1}m

_{2}+ n

_{1}n

_{2}

cos = - = 120

^{0}

__:__

**Straight line**__:__

**Vectorial eq. of line**Let a line passing through a point of P.V. & || to given line EF().

Then, eq. of line

Proof: Let P be any point on line AP & its P.V. is

Then, = = (By law)

__Dumb Question__: Why = ?

Ans: Since is || to EF of P.V.

So, =

__: Eq. of line through origin & || to is =__

**Note**__:__

**Cartesion eq. of straight line**Passing through point & parallel to given vector .

Let line is passing through A(x

_{1}, y

_{1}, z

_{1}) & || to EF whose d.r.’s are a, b, c.

d.r.’s of AP = (x - x

_{1}, y - y

_{1}, z - z

_{1})

d.r.’s of EF = (a, b, c)

Since EF || AP, then,

b Eq. of line in parametric form.

__: Find eq. of line || to & passing through pt.(1, 2, 3) ?__

**Illustration**Ans: A(1, 2, 3) =

eq. of line passing through & || to

= () + ()

__:__

**Vector eq. of line passing through two given points**Vector eq. of line passing through two points whose P.V. S & is:

= + ( - )

Proof: is collinear with

=

__:__

**Cartesian form**Eq. of line passing through (x

_{1}, y

_{1}, z

_{1}) & (x

_{2}, y

^{2}, z

_{2})

D.R.’s of AB = (x

_{2}- x

_{1}, y

_{2}- y

_{1}, z

_{2}- z

_{1})

D.R.’s of AP = (x - x

_{1}, y - y

_{1}, z - z

_{1})

Since AB || AP

__: Find cartesian eq. of line are 6x + 2 = 3y - 1 = 2z + 2. Find its direction ratios.__

**Illustration**Ans: is cartesion eq. of line

6x + 2 = 3y - 1 = 2z + 2

6(x + ) = 3(y - ) = 2(z + 1)

on comparing a = , b = , c =

__:__

**Angle b/w Two lines**=

Angle b/w two lines cos =

If

=

__:__

**Condition for perpendicularity**. = 0 a

_{1}a

_{2}+ b

_{1}b

_{2}+ c

_{1}c

_{2}

__:__

**Condition for parallelism**__: Why angle between pair of lines__

**Dumb Question**Ans: &

__:__

**Perpendicular distance of point froma line**(a)

__:__

**Castesian form**Suppose ‘L’ is foot of line of .

Since L lies on line AB so,

coordinate of L is

=

i.e. L(x

_{1}+ a, y

_{1}+ b, z

_{1}+ c)

direction ratios of PL are:

(x

_{1}+ a - , y

_{1}+ b - , z

_{1}+ c - )

also direction ratios of AB are (a, b, c)

Since PL AB

a(x

_{1}+ a - ) + b(b - ) + c(z

_{1}+ c - ) = 0