Question

If $\bar a,\bar b,\bar c$ are the three non-coplanar vectors, then $\dfrac{{\bar a.\bar b \times \bar c}}{{\bar c \times \bar a.\bar b}} + \dfrac{{\bar b.\bar a \times \bar c}}{{\bar c.\bar a \times \bar b}} = $
A) $0$
B) $2$
C) $ - 2$
D) None of these

Answer 1

Hint: If three vectors are non-coplanar, then, the vector operations between them like $\bar a.\bar b \times \bar c$ can be written as $\left[ {\bar a\bar b\bar c} \right]$, this value is same as, $\bar c \times \bar a.\bar b = \bar a \times \bar b.\bar c = - \left( {\bar b.\bar a \times \bar c} \right)$ etc., or in other words, the terms which follow the cyclic order $\bar a,\bar b,\bar c$ have similar value, and if don’t follow the cyclic order, then tend to have a negative value of $\left[ {\bar a\bar b\bar c} \right]$. So, in this problem, we are to find those with similar values, and calculate the required value.

Complete step by step answer:
Given, $\bar a,\bar b,\bar c$ are three non-coplanar vectors.
We have to find, the value of $\dfrac{{\bar a.\bar b \times \bar c}}{{\bar c \times \bar a.\bar b}} + \dfrac{{\bar b.\bar a \times \bar c}}{{\bar c.\bar a \times \bar b}}$
So, we know, that if three non-coplanar vectors are in the form of $\bar a.\bar b \times \bar c$ or any form following the cyclical order of $\bar a,\bar b,\bar c$ have same value and we write it as $\left[ {\bar a\bar b\bar c} \right]$.
So, the terms of the required vector equation are,
$\bar a.\bar b \times \bar c = \left[ {\bar a\bar b\bar c} \right]$
\[\bar c \times \bar a.\bar b = \left[ {\bar c\bar a\bar b} \right]\]
But, as it follows the same cyclical order, so, the value is equal to $\left[ {\bar a\bar b\bar c} \right]$.
That is, $\bar c \times \bar a.\bar b = \left[ {\bar c\bar a\bar b} \right] = \left[ {\bar a\bar b\bar c} \right]$
Similarly, $\bar c.\bar a \times \bar b = \left[ {\bar c\bar a\bar b} \right] = \left[ {\bar a\bar b\bar c} \right]$
And, $\bar b.\bar a \times \bar c = \left[ {\bar b\bar a\bar c} \right]$
Now, this term doesn’t follow the cyclical order, it’s value will be equal to $ - \left[ {\bar a\bar b\bar c} \right]$
That is, $\bar b.\bar a \times \bar c = \left[ {\bar b\bar a\bar c} \right] = - \left[ {\bar a\bar b\bar c} \right]$
So, substituting the values in the given vector equation, we get,
$\dfrac{{\bar a.\bar b \times \bar c}}{{\bar c \times \bar a.\bar b}} + \dfrac{{\bar b.\bar a \times \bar c}}{{\bar c.\bar a \times \bar b}} = \dfrac{{\left[ {\bar a\bar b\bar c} \right]}}{{\left[ {\bar c\bar a\bar b} \right]}} + \dfrac{{\left[ {\bar b\bar a\bar c} \right]}}{{\left[ {\bar c\bar a\bar b} \right]}}$
Simplifying the expression with the help of known quantities, we get,
$ \Rightarrow \dfrac{{\bar a.\bar b \times \bar c}}{{\bar c \times \bar a.\bar b}} + \dfrac{{\bar b.\bar a \times \bar c}}{{\bar c.\bar a \times \bar b}} = \dfrac{{\left[ {\bar a\bar b\bar c} \right]}}{{\left[ {\bar a\bar b\bar c} \right]}} + \left( {\dfrac{{ - \left[ {\bar a\bar b\bar c} \right]}}{{\left[ {\bar a\bar b\bar c} \right]}}} \right)$
Opening the brackets,
$ \Rightarrow \dfrac{{\bar a.\bar b \times \bar c}}{{\bar c \times \bar a.\bar b}} + \dfrac{{\bar b.\bar a \times \bar c}}{{\bar c.\bar a \times \bar b}} = \dfrac{{\left[ {\bar a\bar b\bar c} \right]}}{{\left[ {\bar a\bar b\bar c} \right]}} - \dfrac{{\left[ {\bar a\bar b\bar c} \right]}}{{\left[ {\bar a\bar b\bar c} \right]}}$
Now, cancelling the like terms in numerator and denominator, we get,
$ \Rightarrow \dfrac{{\bar a.\bar b \times \bar c}}{{\bar c \times \bar a.\bar b}} + \dfrac{{\bar b.\bar a \times \bar c}}{{\bar c.\bar a \times \bar b}} = 1 - 1$
Carrying out the calculations, we get,
$ \Rightarrow \dfrac{{\bar a.\bar b \times \bar c}}{{\bar c \times \bar a.\bar b}} + \dfrac{{\bar b.\bar a \times \bar c}}{{\bar c.\bar a \times \bar b}} = 0$
Therefore, the value of the vector equation $\dfrac{{\bar a.\bar b \times \bar c}}{{\bar c \times \bar a.\bar b}} + \dfrac{{\bar b.\bar a \times \bar c}}{{\bar c.\bar a \times \bar b}}$ is $0$.
Hence, the option (A) is the correct answer.

Note:
The vector operation \[\left[ {\bar a\bar b\bar c} \right]\] is also called a scalar triple product of three vectors. Scalar triple product of three vectors represents the volume of parallelepiped formed by the three vectors. In the case of three non-coplanar vectors, if between the three vectors, at least two of the vectors are similar, then, the value of the resultant vector on operation becomes $0$. That is if, the three vectors are like $\bar a \times \bar c.\bar a$, which we can write as $\left[ {\bar a\bar c\bar a} \right]$, then, the value of the resultant vector is $0$, i.e., $\left[ {\bar a\bar c\bar a} \right] = 0$.