`hat(u)` and `hat(v)` are two non-collinear unit vectos such that `|(hat(u)+hat(v))/(2)+hat(u)xxhat(v)|=1` . Prove that `|hat(u)xxhat(v)|=|(hat(u)-hat(v))/(2)|` .

A plane is at a distance of 5 units from the origin and perpendicular to the vector 2hat(i) + hat(j) + 2hat(k) . The equation of the plane is a) vec(r ).(2hat(i) + hat(j) -2hat(k))=15 b) vec(r ).(2hat(i)+ hat(j)-hat(k))=15 c) vec(r ).(2hat(i) + hat(j)+ 2hat(k))=15 d) vec(r ).(hat(i) + hat(j) + 2hat(k))=15

If 3hat(i) + 2hat(j) -5hat(k)= x(2hat(i) -hat(j) + hat(k)) + y(hat(i) + 3hat(j)-2hat(k))+z(-2hat(i) + hat(j)-3hat(k)) , then