Question

Assertion: If $\theta $ be the angle between $\vec{A}$ and $\vec{B}$ then $\tan \theta =\dfrac{A\times B}{AB}$
Reason: $\vec{A}\times \vec{B}$ is perpendicular to $\vec{A}\vec{B}$.
A. Both assertion and reason are correct and reason is the correct explanation for assertion
B. Both assertion and reason are correct but reason is not the correct explanation for assertion.
C. Assertion is correct but reason is incorrect.
D. Both assertion and reason are incorrect.

Answer 1

Hint: In this question, we are given an assertion and a reason for that assertion. We have to find out whether the reason follows the assertion or not. To find out the correct option, we use the formula of cross product and dot product of two vectors. by putting the formulas and solving the equations, we are able to choose the correct option.

Formula Used:
The formula of Cross product of two vectors is,
$\vec{A}\times \vec{B}=|A|||B|\sin \theta \hat{n}$
Where |A| = length of vector A
And |B| = length of vector B
And, $A.B=$$AB\cos \theta $
Here, $\theta$ is the angle between vectors A and B.

Complete step by step solution:
Given $\theta $ is the angle between $\vec{A}$ and $\vec{B}$. First we must know what is the cross product and dot product of two vectors. Cross product of two vectors is :-
$\vec{A}\times \vec{B}=|A|||B|\sin \theta \,\hat{n}$
Dot product of two vectors is :-
$A.B=$ $AB\cos \theta $
$\dfrac{\vec{A}\times \vec{B}}{\vec{A}.\vec{B}}=\dfrac{AB\sin \theta \hat{n}}{AB\cos \theta } \\ $
$\Rightarrow \dfrac{\vec{A}\times \vec{B}}{\vec{A}.\vec{B}}=\tan \theta \hat{n} \\ $
Where $\hat{n}$ is the unit vector perpendicular to both $\vec{A}$ and $\vec{B}$.
However,
$\dfrac{\vec{A}\times \vec{B}}{\vec{A}.\vec{B}}=\tan \theta $
$\vec{A}\vec{B}$ is a scalar quantity. Hence both the assertion and reason are not correct.

Thus, option D is correct.

Note: We must remember that the dot product of perpendicular vectors is zero and in the cross product , two vectors are zero vectors if both are parallel or opposite to each other. Also note that the result of the dot product is a scalar quantity while the result of the cross product is a vector quantity. This is the major distinction between the two vector cross products.