If A and B are two vectors satisfying the relation

Question:

If $\overrightarrow{\mathrm{A}}$ and $\overrightarrow{\mathrm{B}}$ are two vectors satisfying the relation $\overrightarrow{\mathrm{A}} \cdot \overrightarrow{\mathrm{B}}=|\overrightarrow{\mathrm{A}} \times \overrightarrow{\mathrm{B}}|$. Then the value of $|\overrightarrow{\mathrm{A}}-\overrightarrow{\mathrm{B}}|$ will be :

  1. $\sqrt{\mathrm{A}^{2}+\mathrm{B}^{2}}$

  2. $\sqrt{\mathrm{A}^{2}+\mathrm{B}^{2}+\sqrt{2} \mathrm{AB}}$

  3. $\sqrt{\mathrm{A}^{2}+\mathrm{B}^{2}+2 \mathrm{AB}}$

  4. $\sqrt{\mathrm{A}^{2}+\mathrm{B}^{2}-\sqrt{2} \mathrm{AB}}$


Correct Option: , 4

Solution:

$\overrightarrow{\mathrm{A}} \cdot \overrightarrow{\mathrm{B}}=|\overrightarrow{\mathrm{A}} \times \overrightarrow{\mathrm{B}}|$

$A B \cos \theta=A B \sin \theta \Rightarrow \theta=45^{\circ}$

$|\overrightarrow{\mathrm{A}}-\overrightarrow{\mathrm{B}}|=\sqrt{\mathrm{A}^{2}+\mathrm{B}^{2}-2 \mathrm{AB} \cos 45^{\circ}}$

Hence option (4).

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