Question

Let u and v are unit vectors and w is a vector such that \[u \times v + u = w\,and\,w \times u = v\], then find the value of \[[\vec u\,\vec v\,\vec w] \] ?

Answer 1

Hint: Here the given question is of vector algebra, where we need to solve according to the properties of vector algebra, in order to solve it we know that unit vector are the vector having a magnitude of one, and some properties like commutative and distributive property will also be used here.

Formulae Used:
\[ \Rightarrow [\vec u\,\vec v\,\vec w] = \vec u.(\vec v \times \vec w)\]

Complete step by step answer:
The given question need to be solved with the help of rules related with the vector algebra in order to get the solution for this question, here first we need to expand the given conditions and then try to reach to the asked question, on solving we get:
Given that:
\[
   \Rightarrow \vec u \times \vec v + \vec u = \vec w\,and\,\vec w \times \vec u = \vec v \\
   \Rightarrow \left( {\vec u \times \vec v + \vec u} \right) \times \vec u = \vec w \times \vec u \\
   \Rightarrow \left( {\vec u \times \vec v} \right) \times \vec u + \vec u \times \vec u = \vec v\,(as\,\vec w \times \vec u = \vec v) \\
   \Rightarrow \left( {\vec u.\vec u} \right)\vec v - \left( {\vec v.\vec u} \right)\vec u + \vec u \times \vec u = \vec v \\
 \]
Using, \[\vec u.\vec u = 1,\,and\,\vec u \times \vec u = 0\], since the dot product of the unit vector is one, and the cross product of the unit vector is zero.
\[
   \Rightarrow \left( 1 \right)\vec v - \left( {\vec v.\vec u} \right)\vec u + 0 = \vec v \\
   \Rightarrow \vec v - \vec v = \left( {\vec v.\vec u} \right)\vec u \\
   \Rightarrow \left( {\vec v.\vec u} \right)\vec u = 0 \\
   \Rightarrow \vec u.\vec v = 0\,(as\,\vec u \ne 0)..................................................(eq1) \\
   \Rightarrow [\vec u\,\vec v\,\vec w] = \vec u.(\vec v \times \vec w) \\
   \Rightarrow given\,\vec u \times \vec v + \vec u = \vec w \\
   \Rightarrow \vec u.(\vec v \times \left( {\vec u \times \vec v + \vec u} \right)) \\
   \Rightarrow \vec u.(\vec v \times (\vec u \times \vec v) + \vec v \times \vec u) \\
   \Rightarrow \vec u.((\vec v.\vec v)\vec u - (\vec v.\vec u)\vec v + \vec v \times \vec u) \\
   \Rightarrow \vec u.(|{{\vec v}^2}|\vec u - 0 + \vec v \times \vec u)\,\sin ce\,(\vec v.\vec u = 0,from\,eq1) \\
   \Rightarrow |{{\vec v}^2}|(\vec u.\vec u) + \vec u.(\vec v \times \vec u) \\
 \]
\[
   \Rightarrow |{{\vec v}^2}||\vec u{|^2} + 0 \\
   \Rightarrow 1\,(\sin ce\,|\vec v|,|\vec u| = 1\,unit\,vectors) \\
   \Rightarrow \therefore [\vec u\,\vec v\,\vec w] = 1 \\
 \]
Here we get the answer for the question, the asked magnitude value is one.

Note: Here the given question is of vector algebra and vector algebra works on its properties, multiplication, division has its own rule, here there are two types of multiplication, one is dot product and the other one is cross product and they act separately.