If `lambda(2bar(i)-4bar(j)+4bar(k))` is a unit vector then `lambda=`

If 3bar(i)+3bar(j)+sqrt(3)bar(k),bar(i)+bar(j),sqrt(3)bar(i)+sqrt(3)bar(j)+lambdabar(k) are coplanar vectors then lambda=

If bar(a)=bar(i)+bar(j)+bar(k) , bar(b)=bar(i)+2bar(j)+3bar(k) then a unit vector perpendicular to both vectors (bar(a)+bar(b)) and (bar(a)-bar(b)) is

The volume of a parallelopiped whose edges are represented by -12bar(i)+lambdabar(k) , 3bar(j)-bar(k) and 2bar(i)+bar(j)-15bar(k) is 546 then lambda=

Let bar(a)=2bar(i)+bar(j)+bar(k),bar(b)=bar(i)+2bar(j)-bar(k) and a unit vector bar(c) be coplanar.If bar(c) is perpendicular to bar(a), then bar(c) is

The adjacent sides of a parallelogram are 2bar(i)+4bar(j)-5bar(k) and bar(i)+2bar(j)+3bar(k) then the unit vector parallel to a diagonal is

The orthogonal projection of bar(a)=2bar(i)+3bar(j)+3bar(k) on bar(b)=bar(i)-2bar(j)+bar(k) (where bar(i)*bar(j)*bar(k) are unit vectors along there mutually perpendicular directions is

If (2bar(i)-bar(j)-3bar(k)),B(4bar(i)+bar(j)-bar(k)),C(bar(i)-3bar(j)+2bar(k)) and D(bar(i)-bar(j)+2bar(k)) then the vector equation plane parallel to bar(ABC) and passing the centroid of the tetrahedron ABCD is

If bar(a)=bar(i)+2bar(j)+bar(k) , (b)=bar(i)-bar(j)+bar(k) and bar(c)=bar(i)+bar(j)-bar(k) . A vector in the plane of bar(a) and bar(b) whose projection on bar(c) is (1)/(sqrt(3)) is

The distance between the line bar(r)=2bar(i)-2bar(j)+3bar(k)+lambda((bar(i)-bar(j)+4bar(k)) and the plane bar(r).(bar(i)+5bar(j)+bar(k))=5 is

ABCD is a rhombus.If bar(AC)=bar(i)+(1+lambda)bar(j)+(lambda-2)bar(k) and bar(BD)=(1+lambda)bar(i)+bar(j)-bar(k) then lambda=