Question

If \[\overrightarrow {{\text{u,}}} \overrightarrow {\text{v}} {\text{ }}\] and $\overrightarrow w $ are three non-coplanar vectors, then \[(\overrightarrow u + \overrightarrow v - \overrightarrow w ).(\overrightarrow u - \overrightarrow v ) \times (\overrightarrow v - \overrightarrow w )\] equals
A) $(\overrightarrow {\text{u}} .\overrightarrow {\text{v}} \times \overrightarrow {\text{w}} {\text{)}}$
B) $0$
C) $2\overrightarrow u .(\overrightarrow v \times \overrightarrow w )$
D) $\overrightarrow u .(\overrightarrow w \times \overrightarrow v )$

Answer 1

Hint:
Here we will be using the distributive law in mathematics. It is the main formula in this type of vector question so here we will be using this for deriving the question. And finally, we will get the answer. Here we will use only simplify or elaborating values.

Formula Using:
Distributive law formula = $a(b + c) = ab + ac$

Complete step by step solution:
Using the distributive law, together with the identities
\[\overrightarrow a \times \overrightarrow a \]
Here finally we will this equation $
  \overrightarrow a .(\overrightarrow a \times \overrightarrow b ) = 0 \\
  \overrightarrow b .(\overrightarrow a \times \overrightarrow b ) = 0 \\
 $
So, we will be using above law in our question
\[(\overrightarrow u + \overrightarrow v - \overrightarrow w ).((\overrightarrow u - \overrightarrow v ) \times (\overrightarrow v - \overrightarrow w ))\]
Derived above equation is the same as the distributive law.
$(\overrightarrow u .(\overrightarrow u - \overrightarrow v ) \times (\overrightarrow v - \overrightarrow w )) + ((\overrightarrow v - \overrightarrow w ).((\overrightarrow u - \overrightarrow v ) \times (\overrightarrow v - \overrightarrow w ))$
Simplifying the above equation, we get finally
\[\overrightarrow u .((\overrightarrow u - \overrightarrow v ) \times (\overrightarrow v - \overrightarrow w ))\]
Again, we will simplify the above value
\[\overrightarrow u .((\overrightarrow u \times \overrightarrow v ) - (\overrightarrow u \times \overrightarrow w ) + (\overrightarrow v \times \overrightarrow w ))\]
Elaborate the above equation
\[\overrightarrow u .(\overrightarrow u \times \overrightarrow v ) - \overrightarrow {u.} (\overrightarrow u \times \overrightarrow w ) + \overrightarrow {u.} (\overrightarrow v \times \overrightarrow w )\]
Finally, we will get
$\overrightarrow u .(\overrightarrow v \times \overrightarrow w )$
Here final answer is $\overrightarrow u .(\overrightarrow v \times \overrightarrow w )$
So, here option $d$ is the correct answer.

Additional Information:
Here a non-coplanar vector means not occupying the same surface or linear plane. It is not coplanar two non coplanar points. Coplanar vectors are the vectors which lie on the same plane, in a three-dimensional space. These are vectors which are parallel to the same plane. We can always find in a plane any two random vectors, which are coplanar. Also learn, coplanarity of two lines in a three-dimensional space, represented in vector form.

Note:
Here we will note the coplanar vector or non coplanar vector. If non-coplanar vectors use this method otherwise, we have another method for solving a problem. So, we are solving this type of question and we have an idea about what type of vector we will use.