Question

If a and b are two non - zero, non - collinear vectors, then 2[a b i]i + 2[a b j]j + 2[a b k]k + [a b a] is equal to?
1) 2 (a × b)
2) a × b
3) a + b
4) None of these

Answer 1

Hint: In this type of question, we should know about the “vectors”. Vector is defined as a physical quantity that has both magnitude and direction. It is often represented by an arrow whose length is proportional to the magnitude of the quantity and whose direction is the same as that of the quantity.

Complete step by step Solution:
Right hand rule
Using the right-hand rule, the vector product's direction may be shown. The thumb of your right hand will point in the direction of the vector product if you curl your fingers in a rotation from vector A to vector B.
The product of the magnitudes and the cosine of the smaller angle between the two vectors, A and B, is known as the scalar or dot product of the two vectors.
Properties of dot products and cross products are:
$\overset{\hat{\ }}{\mathop{i}}\,.\overset{\hat{\ }}{\mathop{i}}\,=\overset{\hat{\ }}{\mathop{j}}\,.\overset{\hat{\ }}{\mathop{j}}\,=\overset{\hat{\ }}{\mathop{k}}\,.\overset{\hat{\ }}{\mathop{k}}\,=1$
$\overset{\hat{\ }}{\mathop{i}}\,.\overset{\hat{\ }}{\mathop{j}}\,=\overset{\hat{\ }}{\mathop{j}}\,.\overset{\hat{\ }}{\mathop{k}}\,=\overset{\hat{\ }}{\mathop{k}}\,.\overset{\hat{\ }}{\mathop{i}}\,=0$
$\overset{\hat{\ }}{\mathop{i}}\,\times \overset{\hat{\ }}{\mathop{i}}\,=\overset{\hat{\ }}{\mathop{j}}\,\times \overset{\hat{\ }}{\mathop{j}}\,=\overset{\hat{\ }}{\mathop{k}}\,\times \overset{\hat{\ }}{\mathop{k}}\,=0$
$\overset{\hat{\ }}{\mathop{i}}\,\times \overset{\hat{\ }}{\mathop{j}}\,=\overset{\hat{\ }}{\mathop{k}}\,;\overset{\hat{\ }}{\mathop{j}}\,\times \overset{\hat{\ }}{\mathop{k}}\,=\overset{\hat{\ }}{\mathop{i}}\,;\overset{\hat{\ }}{\mathop{k}}\,\times \overset{\hat{\ }}{\mathop{i}}\,=\overset{\hat{\ }}{\mathop{j}}\,$
Now, we have, 2[a b i]i + 2[a b j]j + 2[a b k]k + [a b a]
Suppose, $\overset{\to }{\mathop{a}}\,={{a}_{1}}i+{{a}_{2}}j+{{a}_{3}}k$
                  $\overset{\to }{\mathop{b}}\,={{b}_{1}}i+{{b}_{2}}j+{{b}_{3}}k$
=\[2\left[ a\text{ }b\text{ }i \right]i\text{ }+\text{ }2\left[ a\text{ }b\text{ }j \right]j\text{ }+\text{ }2\left[ a\text{ }b\text{ }k \right]k\text{ }+\text{ }\left[ a\text{ }b\text{ }a \right]\]
=\[2\left( a\text{ }\times \text{ }b \right)\text{ }i\text{ }.\text{ }i\text{ }+\text{ }2\left( a\text{ }\times \text{ }b \right)\text{ }j\text{ }.\text{ }j\text{ }\text{ }2\left( a\text{ }\times \text{ }b \right)\text{ }k\text{ }.\text{ }k\text{ }+\text{ }0\] $(\because [aab]=0)$
=\[2\left( a\text{ }\times \text{ }b \right)\text{ }+\text{ }2\left( a\text{ }\times \text{ }b \right)\text{ }\text{ }2\left( a\text{ }\times \text{ }b \right)\] ($\overset{\hat{\ }}{\mathop{i}}\,.\overset{\hat{\ }}{\mathop{i}}\,=\overset{\hat{\ }}{\mathop{j}}\,.\overset{\hat{\ }}{\mathop{j}}\,=\overset{\hat{\ }}{\mathop{k}}\,.\overset{\hat{\ }}{\mathop{k}}\,=1$)
=$2(a\times b)$

Therefore, the correct option is (1).

Note: Many real-world scenarios involving force or velocity can be applied to vectors. Take the forces on a boat crossing a river as an illustration. A force is produced in one direction by the boat's motor and another in another by the river's stream.