Q. 5

# Show that the lines and intersect and find their point of intersection.

Given : Cartesian equations of lines  To Find : distance d

Formulae :

1. Equation of line :

Equation of line passing through point A (a1, a2, a3) and having direction ratios (b1, b2, b3) is Where, And 2. Cross Product :

If are two vectors  then, 3. Dot Product :

If are two vectors  then, 4. Shortest distance between two lines :

The shortest distance between the skew lines and is given by, Given Cartesian equations of lines Line L1 is passing through point (0, 2, -3) and has direction ratios (1, 2, 3)

Therefore, vector equation of line L1 is And Line L2 is passing through point (2, 6, 3) and has direction ratios (2, 3, 4)

Therefore, vector equation of line L2 is Now, to calculate distance between the lines,  Here,    Therefore,        Now,  = - 2 + 8 – 6

= 0

Therefore, the shortest distance between the given lines is   As d = 0

Hence, given lines intersect each other.

Now, general point on L1 is

x1 = λ , y1 = 2λ+2 , z1 = 3λ-3

let, P(x1, y1, z1) be point of intersection of two given lines.

Therefore, point P satisfies equation of line L2.  3λ – 6 = 4λ – 8

λ = 2

Therefore, x1 = 2 , y1 = 2(2)+2 , z1 = 3(2)-3

x1 = 2 , y1 = 6 , z1 = 3

Hence point of intersection of given lines is (2, 6, 3).

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