Find the shortest distance between the straight lines
`vecr = -4hati+ 4 hat j+ hat k+lambda_1(hat i+hatj-hatk)` and `vecr = -3hati-8hatj+3hatk+lambda_2(hati+3hatj-3hatk)`.

Find the shortest distance between the straight lines vecr=-4hati+4hatj+hatk+lambda_1(hati+hatj-hatk)" and "vecr=-3hati-8hatj-3hatk+lambda_2(2hati+3hatj+3hatk) .

The shortest distance between the lines whose vector equations are vecr=hati-2hatj+3hatk+lambda(hati-hatj+hatk) and vecr=hati-hatj-hatk+mu(hati+2hatj-2hatk) is -

The angle between the line vecr=( hati+2 hatj- hatk)+ lambda( hati- hatj+ hatk) and the plane vecr. (2 hati- hatj+ hatk)=4 is __

Find the angle between the lines vecr = (2hati - hatj +3hatk) + lambda(hati + hatj + 2hatk) and vec r = (hati - 3hatj)+mu(2hatj - hatk)

The cosine of the angle between the lines vecr=3hati+2hatj-4hatk+lambda(hati+2hatj+2hatk) and vecr=5hati-2hatj+mu(3hati+2hatj+6hatk) is -

Find the distance between the parallel planes vecr.(2 hati-3 hatj+ 6hatk)=5 and vecr .(6 hati-9 hatj+ 18hatk)+20=0 .

Find the equation of a line passing through the point P(2,-1,3) and perpendicular to the lines vecr=(hati+hatj-hatk)+lambda(2hati-2hatj+hatk) and vec r=(2hati-hatj-3hatk)+mu(hati+2hatj+2hatk) .

Find the vector and cartesion equations of a plane containing the two lines vec r=2hatj+hatj-3hatk+lambda(hati+2hatj+5hatk) and vec r=3hati+3hatj+2hatk+mu(3hati-2hatj+5hatk) .Also show that the line vec r=2hati+5hatj+2hatk+p(3hati-2hatj+5hatk) lies in the plane.

Find the angle between the two vectors vecA = hati + 2hatj + 3hatk and vecB = 2hati+ hatj + 4hatk .

Find the angle between the plane vecr.(2 hati-hatj+2 hat k)=6 and vecr.(3 hati+6 hatj-2 hatk) =9