Question

If\[\left| {A \times B} \right| = \sqrt 3 A \cdot B\], then the value of \[\left| {A \times B} \right|\]is:
A. \[{\left( {{A^2} + {B^2} + AB} \right)^{\dfrac{1}{2}}}\]
B. \[{\left( {{A^2} + {B^2} + \dfrac{{AB}}{{\sqrt 3 }}} \right)^{\dfrac{1}{2}}}\]
C. A +B
D. \[{\left( {{A^2} + {B^2} + \sqrt 3 AB} \right)^{\dfrac{1}{2}}}\]

Answer 1

Hint:

a. You should know vector calculus.
b. You should know vector identities.
c. You should know cosine values for \[0,30,{\text{ }}45,{\text{ }}60,{\text{ }}and{\text{ }}90\].


Complete step by step solution:


We know that, by vector product,
\[\left( {A \times B} \right) = AB\sin \theta \hat n\] - - - - - - - - - - - - - - - - - - - - (1)


And by Scalar product,
\[\left( {A \cdot B} \right) = AB\cos \theta \]- - - - - - - - - - - - - - - - - - - - - - (2)
In question, the given quantities are
\[\left| {A \times B} \right| = \sqrt 3 A \cdot B\]- - - - - - - - - - - - - - - - - - - - - - -(3)
By using
\[\left| {A \times B} \right| = \left| A \right|\left| B \right|\sin \theta \] - - - - - - - - - - - - - - - - - - - - - (4)
We get,
\[\left| {A \times B} \right| = AB\sin \theta \] - - - - - - - - - - - - - - - - - - - - - - - (5)
Now,
\[\left| {A \cdot B} \right| = \left| A \right|\left| B \right|\cos \theta \]
\[A \cdot B = AB\cos \theta \] - - - - - - - - - - - - - - - - - - - - - - - - - (6)
Substitute equation (6) in the right side of the equation (3) and equation (5) in the left side of the equation (3).
So,
\[AB\sin \theta = \sqrt 3 AB\cos \theta \]
\[\dfrac{{\sin \theta }}{{\cos \theta }} = \sqrt 3 \]
\[\tan \theta \]\[ = \] \[\sqrt 3 \]
\[ \Rightarrow \theta = 60^\circ \]
Now,
\[{(A + B)^2} = {A^2} + {B^2} + 2A.B\]
\[{(A + B)^2} = {A^2} + {B^2} + 2AB\cos \theta \]
\[{(A + B)^2} = {A^2} + {B^2} + 2AB \cdot \dfrac{1}{2}\]
\[{(A + B)^2} = {A^2} + {B^2} + AB\]
Or,
\[A + B = {\left( {{A^2} + {B^2} + AB} \right)^{\dfrac{1}{2}}}\]

Hence, Option (A) is correct

Note:
a. Substitution should not be reversed. If it reverses, all calculation will be wasted and time too.
b. Should be aware of the values of cos, sin and tan for \[0,{\text{ }}30,{\text{ }}45,{\text{ }}60,{\text{ }}and{\text{ }}90.\]
c. After getting the final calculation check the options and conveniently rearrange the solution.