If $a(\alpha \times \beta)+b(\beta \times \gamma)+c(\gamma \times \alpha)=0$ and at least one of the scalars $a, b, c$ is non-zero,then the vectors $\alpha, \beta, \gamma$ are

Three vectors $\hat{i}-\hat{k}$,$\lambda \hat{i}+\hat{j}+(1-\lambda) \hat{k}$,and $\mu \hat{i}+\lambda \hat{j}+(1+\lambda-\mu) \hat{k}$ represent the coterminous edges of a parallelepiped. The volume of the parallelepiped depends on:

If $a, b, c$ are three coplanar vectors,then $[a + b, b + c, c + a] = $

Statement-$1$: Vectors $\vec{a}, \vec{b},$ and $\vec{c}$ are coplanar if and only if $\vec{a} \cdot (\vec{b} \times \vec{c}) = 0$.
Statement-$2$: Vectors $\vec{u}$ and $\vec{v}$ are perpendicular if and only if $\vec{u} \cdot \vec{v} = 0$,where $\vec{u} \times \vec{v}$ is a vector perpendicular to the plane of $\vec{u}$ and $\vec{v}$.

If the vectors $2i + j - k$,$-i + 2j + \lambda k$,and $-5i + 2j - k$ are coplanar,then the value of $\lambda$ is equal to:

If the vectors $m \hat{i} + m \hat{j} + n \hat{k}$,$\hat{i} + \hat{k}$,and $n \hat{i} + n \hat{j} + p \hat{k}$ lie in a plane,then...