The volume of the tetrahedron whose vertices are the points $\hat i,\hat i+\hat j, \hat i+\hat j+\hat k$ and $2\hat i+3\hat j +\lambda \hat k$ is 1/6 units. Then, the values of λ

is any real value

Let ABCD be the given tetrahedron. Then,

$\vec{AB}=\hat j, \vec{AC} =\hat j +\hat k$ and $\vec{AD} =\hat i +3\hat j +λ\hat k$

∴ Volume = $\frac{1}{6}$

$⇒\frac{1}{6}[\vec{AB}\,\,\vec{AC}\,\,\vec{AD}]=\frac{1}{6}$

$⇒[\vec{AB}\,\,\vec{AC}\,\,\vec{AD}]=1$

$⇒(\vec{AB}×\vec{AC}).\vec{AD}=1$

$⇒\hat i.(\hat i+3\hat j+λ\hat k)=1$, which is true for all values of λ.