Obtain the distance betweenn two parallel lines `vec(r)=vec(a)_(1)+lambdavec(b), vec(r)=vec(a)_(2)+muvec(b)`.

Derive the formula for the distance between two parallel lines vec(r)=vec(a_(1))+lambdavec(b) and vec(r)=vec(a_(2))+muvec(b) in vector form.

If |vec(a) + vec(b)| = |vec(a) - vec(b)| then

Show that (vec(a)-vec(b))xx(vec(a)+vec(b))=2 (vec(a)xxvec(b)) .

Show that the lines vec(r)= vec(i)+vec(j)-vec(k)+lambda(3 vec(i)- vec(j)) and vec(r)= 4 vec(i)-vec(k)+ mu (2 vec(i) +3 vec(k)) intersect. Also find their point of intersection.

Prove that [vec(a)+vec(b),vec(b)+vec( c) , vec( c) +vec(a)] =2 [vec(a),vec(b),vec(c )] .

Find |vec(b)| , if (vec(a) + vec(b)).(vec(a) -vec(b)) = 8 and |vec(a)| = 8|vec(b)|

Prove that [vec(a),vec(b),vec( c) +vec(d)]=[vec(a),vec(b),vec( c)] +[vec(a),vec(b),vec(d)] .