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Show that lines: r=i+j+k+λ(i-hat+k) r=4j+2k+μ(2i-j+3k) are coplanar  Also, find the equation of the plane containing these lines. - Mathematics

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Show that lines: 

`vecr=hati+hatj+hatk+lambda(hati-hat+hatk)`

`vecr=4hatj+2hatk+mu(2hati-hatj+3hatk)` are coplanar 

Also, find the equation of the plane containing these lines.

 
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Solution

 
 

`vecr=hati+hatj+hatk+lambda(hati-hat+hatk) .......(i)`

Convert ing into cartesian form,

`(x-1)/1=(y-1)/-1=(z-1)/1`

(x1,y1,z1)=(1,1,1)

a1=1,b1=-1,c1=1

`vecr=4hatj+2hatk+mu(2hati-hatj+3hatk)......(ii)`

Convert ing into cartesian form,

(x2,y2,z2)=(0,4,2)

a2=2,b2=-1,c2=3

Condition for the lines to be coplanar is

`|[0-1,4-1,2-1],[1,-1,1],[2,-1,3]|=|[-1,3,1],[1,-1,1],[2,-1,3]|=0`

the lines are coplanar

Intersection of the two lines is
Let the equation be a(x-x1 )+b(y -y1 )+ c(z - z1 )= 0.....(iii)

Direction ratios of the plane is

a-b+c=0

2a-b+3c=0

Solving by cross-multiplication

`a/(-3+1)=b/(2-3)=c/(-1+2)`

Since the plane passes through (0,4,2) from line (ii)

a(x-0)+b(y-4)+c(z-2)=0

`=>-2lambdax-lambda(y-4)+lambda(z-2)=0`

`=>-2x-y+4+z-2=0`

`=>-2x-y+z=-2`

`=>2x+y-z=2`

 
 
Concept: Shortest Distance Between Two Lines
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