Show that the lines `vec(r)=(6hat(i)+hat(j)+2hat(k))+s(hat(i)+2hat(j)-3hat(k)),andvec(r)=(3hat(i)+2hat(j)-2hat(k))+t(2hat(i)+4hat(j)-5hat(k))` are skew lines and hence find the shortest distance between them.

Show that the straight lines vec(r)=(5hat(i)+7hat(j)-3hat(k))+s(4hat(i)+4hatj-5hat(k))andvec(r)=(8hat(i)+4hat(j)+5hat(k))+t(7hat(i)+hat(j)+hat(k)) are coplanar. Find the vector equation of the plane in which they lie.

Find the acute angle between the following lines. vec(r)=(4hat(i)-hat(j))+t(hat(i)+2hat(j)-2hat(k)) vec(r)=(hat(i)-2hat(j)+4hat(k))+s(-hat(i)-2hat(j)+2hat(k))

Find the angle between the line vec(r)=(2hat(i)-hat(j)+hat(k))+t(6hat(i)+2hat(j)-2hat(k))" and the plane "vec(r)*(6hat(i)+3hat(j)+2hat(k))=8

Show that the vectors are coplanar hat(i)-2hat(j)+3hat(k),-2hat(i)+3hat(j)-4hat(k),-hat(j)+2hat(k)

The angle between the lines vec(r)=(hat(i)+2hat(j)-3hat(k))+t(2hat(i)+hat(j)-2hat(k))" and the plane "vec(r)*(hat(i)+hat(j))+4=0 is

Show that the vectors are coplanar 2hat(i)+3hat(j)+hat(k),hat(i)-hat(j),7hat(i)+3hat(j)+2 hat(k)

Show that the vectors 2hat(i)-hat(j)+hat(k),3hat(i)-4hat(j)-4hat(k), hat(i)-3hat(j)-5hat(k) form a right angled triangle.

show that the vectors 3 hat(i)- 2hat(j)+ hat(k), hat(i)-3hat(j)+5hat(k) and 2hat(i)+ hat(j)- 4 hat(k) form a right angled triangle