Question

$\vec A$, $\vec B$ and $\vec C$ are three non-collinear, non-co-planar vectors. What can you say about the direction of $\vec A \times \left( {\vec B \times \vec C} \right)$?

Answer 1

The resultant vector of a cross product of two vectors is always perpendicular to the plane containing the two vectors. The direction of the resultant vector can be determined by using the Right hand thumb rule.

Complete step by step answer:
It is given that $\vec A$, $\vec B$ and $\vec C$ are non-collinear and non-coplanar vectors.
It is required to find the direction of $\vec A \times \left( {\vec B \times \vec C} \right)$.
Let ${\vec R_{ABC}} = \vec A \times \left( {\vec B \times \vec C} \right)$
And ${\vec R_{BC}} = \vec B \times \vec C$
Apply the right-hand thumb rule for $\vec B \times \vec C$. The direction of ${\vec R_{BC}}$ will be perpendicular to the plane containing the vectors $\vec B$ and $\vec C$.
Now ${\vec R_{ABC}} = \vec A \times {\vec R_{BC}}$
Similarly, the direction of ${\vec R_{ABC}}$ will be perpendicular to the plane containing $\vec A$ and ${\vec R_{BC}}$.
Since, ${\vec R_{BC}}$ is perpendicular to the plane containing $\vec B$ and $\vec C$, The resultant vector will be lie on the plane containing $\vec B$ and $\vec C$.
Hence, $\vec A \times \left( {\vec B \times \vec C} \right)$ is perpendicular to the vector $\vec A$ and lie on the plane containing $\vec B$ and $\vec C$.

Note:
A vector is a physical quantity which has both magnitude and direction. The vector product of two vectors $\vec A$ and $\vec B$ is $\vec R = \vec A \times \vec B = \left| {\vec A} \right|\left| {\vec B} \right|\sin \theta {\text{ }}\widehat n$. Where $\theta$ is the angle between $\vec A$ and $\vec B$. $\widehat n$ is the unit vector perpendicular to the plane containing $\vec A$ and $\vec B$.
Three vectors in 3D are said to be coplanar if their scalar triple product is zero.