एक m द्रव्यमान का पिण्ड बल vec F = (3hati +2hatj + 4hatk)N के प्रभाव में बिन्दु A(3, 1, 2) से बि...

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 )

Find the shortest distance between the lines: (i) vec(r) = 3 hati + 8 hat(j) + 3 hatk + lambda (3 hati - hatj + hatk) and vec(r) = - 3 hat(i) - 7 hatj + 6 hatk + mu (-3 hati + 2 hatj + 4 hatk ) (ii) ( hati - hatj + 2 hatk) + lambda ( -2 hati + hatj + 3 hatk ) and (2 hati + 3 hatj - hatk) + mu (3 hati - 2 hatj + 2 hatk). (iii) vec(r) = (hati + 2 hatj + 3 hatk) + lambda ( hati - 3 hatj + 2 hatk ) and vec(r) = (4 hati + 5 hatj + 6 hatk) + mu (2 hati + 3 hatj + hatk) .

Find the shortest distance between the lines whose vector equations are : vec(r) = (hati + 2 hatj + 3 hatk ) + lambda (hati -3 hatj + 2 hatk) and vec(r) = 4 hati + 5 hatj + 6 hatk + mu (2 hati + 3 hatj + hatk) .

If vec(A) = 2hati +3 hatj +hatk and vec(B) = 3hati + 2hatj + 4hatk , then find the value of (vec(A) +vec(B)) xx (vec(A) - vec(B))

Angle between the planes: (i) vec(r). (hati - 2 hatj - hatk) = 1 and vec(r). (3 hati - 6 hatj + 2 hatk) = 0 (ii) vec(r). (2 hati + 2 hatj - 3 hatk ) = 5 and vec(r) . ( 3 hati - 3 hatj + 5 hatk ) = 3

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) .

Show that the lines : vec(r) = 3 hati + 2 hatj - 4 hatk + lambda (hati + 2 hatj + 2 hatk) and vec(r) = 5 hati - 2 hatj + mu (3 hati + 2 hatj + 6 hatk) (ii) vec(r) = (hati + hatj - hatk) + lambda (3 hati - hatj) . and vec(r) = (4 hati - hatk) + mu (2 hati + 3 hatk) are intersecting. Hence, find their point of intersection.

Find the vector and cartesian equations of the plane containing the lines : 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 hatj + 3 hatj + 8 hatk) .

Find the Cartesian equation of the following planes : a. vec(r). (hati + hatj - hatk ) = 2 b. vec(r). (2 hati + 3 hatj - 4 hatk ) = 1 (c ) vec(r). [ (s - 2t) hati + (3 - t ) hatj + (2 s + t ) hatk] = 15