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ConceptShortest Distance Between Two Lines

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#### Question

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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If the lines

`(x-1)/-3=(y-2)/(2k)=(z-3)/2 and (x-1)/(3k)=(y-5)/1=(z-6)/-5`

are at right angle then find the value of k

Show that the four points A(4,5,1), B(0,-1,-1), C(3,9,4) and D(-4,4,4) are coplanar.

Show that the following two lines are coplanar:

`(x−a+d)/(α−δ)= (y−a)/α=(z−a−d)/(α+δ) and (x−b+c)/(β−γ)=(y−b)/β=(z−b−c)/(β+γ)`

Find the shortest distance between the lines

`bar r = (4 hat i - hat j) + lambda(hat i + 2 hat j - 3 hat k)`

and

`bar r = (hat i - hat j + 2 hat k) + mu(hat i + 4 hat j -5 hat k)`

where λ and μ are parameters

#### Reference Material

Solution for concept: Shortest Distance Between Two Lines. For the courses 12th CBSE (Arts), 12th CBSE (Commerce), 12th CBSE (Science)