Vector equations of a line if `vecr=3hati-6hatj+7hatk+lambda(2hati-hatj+hatk)`. Find te equations of the line in the symmetrical form.

A set of D.R of the line vecr=(hati+hatj+hatk)+1(2hati+3hatj+6hatk) are

Show that the line L whose vector equation is vecr=2hati-2hatj+3hatk+lambda(hati-hatj+4hatk) is parallel to the plane pi whose vector equation is vecr*(hati+5hatj+hatk)=5 and find the distance between them.

Write the equation of the following line vecr=hati+hatj+hatk+t(2hati-3hatj+6hatk) , t in R. in the cartestian form.

Show that the line , vecr=2hati-3hatj+5hatk+lambda(hati-hatj+2hatk) lies in the plane vecr*(3hati+hatj-hatk)+2=0 .

Show that the line , vecr=hati+hatj+lambda(2hati+hatj+4hatk) lies in the plane vecr*(hati+2hatj-hatk)=3 .

Find the distance of the point (2, 3, -1) from the point of intersection of the line vecr=3hati+hatj-2hatk+lambda(3hati+6hatj+2hatk) and the plane vecr.(2hati-hatj+hatk) =3.

Find the point of intersection of the line : vecr=(hati+2hatj+3hatk)+lambda(2hati+hatj+2hatk) and the plane vecr*(2hati-6hatj+3hatk)+5=0 .

Find the angle between the line : vecr=(hati-hatj+hatk)+lambda(2hati-hatj+3hatk) and the plane vecr*(2hati+hatj-hatk)=4 . Also find whether the line is parallel to the plane or not.

Find the vector equation of the line passing through (1, 2, 3) and parallel to each of the planes vecr = (hati-hatj+2 hatk)= 5" and "vecr *(3hati- hatj+hatk)= 6 . Also find the point of intersection of the line thus obtained with the plane vecr * (2hati-hatj+hatk)= 4 .

Find the shortest distance between the following lines whose vector equations are : vecr=3hati-15hatj+9hatk+lambda(2hati-7hatj+5hatk) and vecr=(2mu-1)hati+(1+mu)hatj+(9-3mu)hatk