The value of $\left[\overrightarrow{a}-\overrightarrow{b}\overrightarrow{b}-\overrightarrow{c}\overrightarrow{c}-\overrightarrow{a}\right]$ is ?

The correct option is A

$0$

Explanation for the correct option:

We can write $\left[\overrightarrow{a}-\overrightarrow{b}\overrightarrow{b}-\overrightarrow{c}\overrightarrow{c}-\overrightarrow{a}\right]$ as $\left(\overrightarrow{a}-\overrightarrow{b}\right).\left(\left(\overrightarrow{b}-\overrightarrow{c}\right)\times \left(\overrightarrow{c}-\overrightarrow{a}\right)\right)$

Now to solve it we will do the cross product of two vectors $\overrightarrow{b}-\overrightarrow{c}$ and $\overrightarrow{c}-\overrightarrow{a}.$

$$\begin{gathered} =\left(\overrightarrow{a}-\overrightarrow{b}\right)·(\overrightarrow{b}\times \overrightarrow{c}-\overrightarrow{b}\times \overrightarrow{a}-\overrightarrow{c}\times \overrightarrow{c}+\overrightarrow{c}\times \overrightarrow{a}) \\ =\left(\overrightarrow{a}-\overrightarrow{b}\right)·(\overrightarrow{b}\times \overrightarrow{c}-\overrightarrow{b}\times \overrightarrow{a}+\overrightarrow{c}\times \overrightarrow{a})\mathbf{[}\mathbf{\because }\overrightarrow{\mathbf{c}}\mathbf{\times }\overrightarrow{\mathbf{c}}\mathbf{=}\mathbf{0}\mathbf{]} \\ =\left(\overrightarrow{a}-\overrightarrow{b}\right)·(\overrightarrow{b}\times \overrightarrow{c}+\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{c}\times \overrightarrow{a}) \end{gathered}$$

Now we will be finding the dot product of all the terms inside the bracket with $\overrightarrow{a}-\overrightarrow{b}.$

$=\overrightarrow{a}·\overrightarrow{b}\times \overrightarrow{c}+\overrightarrow{a}·\overrightarrow{a}\times \overrightarrow{c}+\overrightarrow{a}·\overrightarrow{c}\times \overrightarrow{a}-\overrightarrow{b}·\overrightarrow{b}\times \overrightarrow{c}-\overrightarrow{b}·\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{b}·\overrightarrow{c}\times \overrightarrow{a}$

Now we know that $\overrightarrow{a}\times \overrightarrow{c}$ is perpendicular to $\overrightarrow{a},\overrightarrow{b}\times \overrightarrow{c}$ is perpendicular to $\overrightarrow{b}$and $\overrightarrow{a}\times \overrightarrow{b}$ is perpendicular to $\overrightarrow{b}$ and taking dot products of the perpendicular vectors gives $\overrightarrow{0}.$

$\Rightarrow \overrightarrow{a}·\overrightarrow{b}\times \overrightarrow{c}-\overrightarrow{b}·\overrightarrow{c}\times \overrightarrow{a}$

We can rewrite it as

$\Rightarrow \overrightarrow{a}·\overrightarrow{b}\times \overrightarrow{c}-\overrightarrow{a}·\overrightarrow{b}\times \overrightarrow{c}=0$

Therefore, $\left[\overrightarrow{a}-\overrightarrow{b}\overrightarrow{b}-\overrightarrow{c}\overrightarrow{c}-\overrightarrow{a}\right]$ is equal to $0.$

Hence, the correct answer is option (A).