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3-1.Vectors
medium
If $\vec A,\vec B$ and $\vec C$ are vectors having a unit magnitude. If $\vec A + \vec B + \vec C = \vec 0$ then $\vec A.\vec B + \vec B.\vec C + \vec C.\vec A$ will be
A
$1$
B
$ - 1.5$
C
$ -0.5$
D
$0$
Solution
$\overrightarrow{\mathrm{A}} \cdot \overrightarrow{\mathrm{B}}+\overrightarrow{\mathrm{B}} \cdot \overrightarrow{\mathrm{C}}+\overrightarrow{\mathrm{C}} \cdot \overrightarrow{\mathrm{A}}$
$=\mathrm{AB} \cos 120^{\circ}+\mathrm{BC} \cos 120^{\circ}+\mathrm{AC} \cos 120^{\circ}$
as $A=B=C=1$
$\overrightarrow{\mathrm{A}} \cdot \overrightarrow{\mathrm{B}}+\overrightarrow{\mathrm{B}} \cdot \overrightarrow{\mathrm{C}}+\overrightarrow{\mathrm{C}} \cdot \overrightarrow{\mathrm{A}}=3 \cos 120^{\circ}=-3 / 2$
Standard 11
Physics
