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Definition of Functions

If the vector...

Question

If the vector $a, b$ and $c$ form the sides $B C, C A$ and $A B$ and equal magnitute respectively of a triangle $A B C,$ then

A

a \cdot b + b \cdot c + c \cdot a = 0

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B

a \times b = b \times c = c \times a

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C

a \cdot b = b \cdot c = c \cdot a

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D

a \times b + b \times c + c \times a = O

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Solution

The correct option is D $a \times b = b \times c = c \times a$
By triangle law, $\to a + \to b + \to c = \to 0$
Taking cross product by $\to a, \to b, \to c$ respectively
$\to a \times (\to a + \to b + \to c) = \to a \times \to 0 = \to 0$
$\Rightarrow \to a \times \to a \times \to a \times \to b + \to a \times \to c = \to a$
$\Rightarrow \to a \times \to b = \to c \times \to a [∵ \to a \times \to a = \to 0]$
Similarly, $\to a \times \to b = \to b \times \to c .$
$∴ \to a \times \to b = \to b \times \to c = \to c \times \to a .$

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