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3-1.Vectors
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If $\vec{A}$ and $\vec{B}$ are two vectors satisfying the relation $\vec{A} . \vec{B}=[\vec{A} \times \vec{B}]$. Then the value of $[\vec{A}-\vec{B}]$. will be :
A
$\sqrt{A^{2}+B^{2}-\sqrt{2} A B}$
B
$\sqrt{A^{2}+B^{2}}$
C
$\sqrt{A^{2}+B^{2}+\sqrt{2} A B}$
D
$\sqrt{A^{2}+B^{2}+\sqrt{2} A B}$
(JEE MAIN-2021)
Solution
$\vec{A} \vec{B}=|\vec{A} \times \vec{B}|$
$A B \cos \theta=A B \sin \theta \Rightarrow \theta=45^{\circ}$
$|\vec{A}-\vec{B}|=\sqrt{A^{2}+B^{2}-2 A B \cos 45^{\circ}}$
$=\sqrt{A^{2}+B^{2}-\sqrt{2} A B}$
Standard 11
Physics
Similar Questions
If $\left| {\vec A } \right|\, = \,2$ and $\left| {\vec B } \right|\, = \,4$ then match the relation in Column $-I$ with the angle $\theta $ between $\vec A$ and $\vec B$ in Column $-II$.
| Column $-I$ | Column $-II$ |
| $(a)$ $\left| {\vec A \, \times \,\,\vec B } \right|\, = \,\,0$ | $(i)$ $\theta = \,{30^o}$ |
| $(b)$ $\left| {\vec A \, \times \,\,\vec B } \right|\, = \,\,8$ | $(ii)$ $\theta = \,{45^o}$ |
| $(c)$ $\left| {\vec A \, \times \,\,\vec B } \right|\, = \,\,4$ | $(iii)$ $\theta = \,{90^o}$ |
| $(d)$ $\left| {\vec A \, \times \,\,\vec B } \right|\, = \,\,4\sqrt 2$ | $(iv)$ $\theta = \,{0^o}$ |
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If $\theta$ is the angle between two vectors $A$ and $B$, then match the following two columns.
| colum $I$ | colum $II$ |
| $(A)$ $A \cdot B =| A \times B |$ | $(p)$ $\theta=90^{\circ}$ |
| $(B)$ $A \cdot B = B ^2$ | $(q)$ $\theta=0^{\circ}$ or $180^{\circ}$ |
| $(C)$ $|A+B|=|A-B|$ | $(r)$ $A=B$ |
| $(D)$ $|A \times B|=A B$ | $(s)$ None |
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