Question

$u = \widehat i \times \left( {\overrightarrow a \times \widehat i} \right) + \widehat j \times \left( {\overrightarrow a \times \widehat j} \right) + \widehat k \times \left( {\overrightarrow a \times \widehat k} \right)$ is equal to?
1. $\overrightarrow a $
2. $2\overrightarrow a $
3. $3\overrightarrow a $
4. None of these

Answer 1

Hint: In this question we are given that $u = \widehat i \times \left( {\overrightarrow a \times \widehat i} \right) + \widehat j \times \left( {\overrightarrow a \times \widehat j} \right) + \widehat k \times \left( {\overrightarrow a \times \widehat k} \right)$ and we have to simplify the expression and find the simplest value. First step is to assume $\overrightarrow a = p\widehat i + q\widehat j + r\widehat k$. Now as it is visible we have to use the cross product. Simply start multiplying the terms but using the formula carefully and you will the answer in terms of vector only.

Formula Used:
Cross product of unit vectors –

$\widehat i \times \widehat j = \widehat k$	$\widehat j \times \widehat i = - \widehat k$
$\widehat j \times \widehat k = \widehat i$	$\widehat k \times \widehat j = - \widehat i$
$\widehat k \times \widehat i = \widehat j$	$\widehat i \times \widehat k = - \widehat j$

Complete step by step Solution:
Given that,
$u = \widehat i \times \left( {\overrightarrow a \times \widehat i} \right) + \widehat j \times \left( {\overrightarrow a \times \widehat j} \right) + \widehat k \times \left( {\overrightarrow a \times \widehat k} \right) - - - - - \left( 1 \right)$
Let $\overrightarrow a = p\widehat i + q\widehat j + r\widehat k$
Put the above value of vector in equation (1),
\[u = \widehat i \times \left( {\left( {p\widehat i + q\widehat j + r\widehat k} \right) \times \widehat i} \right) + \widehat j \times \left( {\left( {p\widehat i + q\widehat j + r\widehat k} \right) \times \widehat j} \right) + \widehat k \times \left( {\left( {p\widehat i + q\widehat j + r\widehat k} \right) \times \widehat k} \right)\]
Using vectors cross product formula,
\[u = \widehat i \times \left( { - q\widehat k + r\widehat j} \right) + \widehat j \times \left( {p\widehat k - r\widehat i} \right) + \widehat k \times \left( { - p\widehat j + q\widehat i} \right)\]
\[u = q\widehat j + r\widehat k + p\widehat i + r\widehat k + p\widehat i + q\widehat j\]
\[u = 2p\widehat i + 2q\widehat j + 2r\widehat k\]
\[u = 2\left( {p\widehat i + q\widehat j + r\widehat k} \right)\]
\[u = 2\overrightarrow a \]

Hence, the correct option is 2.

Note: The key concept involved in solving this problem is a good knowledge of vectors. Students must know that there are two ways to simplify vectors cross-product and dot-product. The cross product is typically used to determine the vector that is perpendicular to the plane surface spanned by two vectors, whereas the dot product is typically used to determine the angle between two vectors or the length of the vector. Here in this question, we have used cross-product because in the question it was given (cross symbol for the product). If the dot symbol will be there we’ll apply the formulas of the dot product of unit vectors. In the dot product, the multiplication of the same unit vectors is one and of a different unit, vectors are zero.