`(4_1-25) r=[3hati-7hatj+2hatk]=15, r=[4hati+3hatj-5hatk]=16`

Find the shortest distance between the lines bar(r) = (4hati - hatj) + lamda(hati + 2hatj - 3hatk) and bar(r) = (hati - hatj + 2hatk) + mu(hati + 4hatj - 5hatk).

Find the shortest distance between the lines: (i) vec(r) = 6 hat(i) + 2 hat(j) + 2 hatk + lambda (hati - 2hatj + 2 hatk) and vec(r) = - 4 hati - hatk + mu (3 hati - 2 hatj - 2 hatk ) (ii) vec(r) = (4 hat(i) - hat(j)) + lambda (hati + 2hatj - 3 hatk) and vec(r) = (hati - hatj + 2hatk) + mu (2 hati + 4 hatj - 5 hatk ) (iii) vec(r) = (hati + 2 hatj - 4 hatk) + lambda (2 hati + 3 hatj + 6 hatk ) and vec(r) = (3 hati + 3 hatj + 5 hatk) + mu (-2 hati + 3 hatj + 6 hatk )

vecr=3hati+2hatj-5hatk, veca=2hati-hatj+hatk, vecb=hati+3hatj-2hatk, vecc=-2hati+hatj-3hatk such that vecr=lambdaveca+muvecb+gammavecc , then

vecr=3hati+2hatj-5hatk, veca=2hati-hatj+hatk, vecb=hati+3hatj-2hatk, vecc=-2hati+hatj-3hatk such that vecr=lambdaveca+muvecb+gammavecc , then

Find the angle between the following pairs of lines : (i) vec(r) = 2 hati - 5 hatj + hatk + lambda (3 hati + 2 hatj + 6 hatk ) and vec(r) = 7 hati - 6 hatk + mu (hati + 2 hatj + 2 hatk) (ii) vec(r) = 3 hati + hatj - 2 hatk + lambda (hati - hatj - 2 hatk ) and vec(r) = 2 hati - hatj - 56 hatk + mu (3 hati - 5 hatj - 4 hatk) .

Find the shortest distance between the lines vecr=(4hati-hatj)+lambda(hati+2hatj-3hatk) and vecr=(hati-hatj+2hatk)+mu(2hati+4hatj-5hatk) .

Find the angle between the vector vecr_(1)=(4hati-3hatj+5hatk) andvecr_(2)=(3hati+4hatj+5hatk) .

Verify if the point having positions vector 4 hati-11 hatj +2 hatk lies on the line overset (-)r =( 6 hati -4 hatj +5hatk ) +mu (2hati +7hatj +3hatk )

Shortest distance between the lines: vecr=(4hati-hatj)+lambda(hati+2hatj-3hatk) and vecr=(hati-hatj+2hatk)+u(2hati+4hatj-5hatk)