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Multiplication of a Vector by a Scalar

Unit vector p...

Question

Unit vector perpendicular to the plane of the triangle $A B C$ with position vectors of the vertices $A, B, C,$ is $($ where $Δ$ is the area of the triangle $A B C$ ) .

A

\frac{(\to a \times \to b + \to b \times \to c + \to c \times \to a)}{Δ}

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B

\frac{(\to a \times \to b + ¯ ¯ b \times \to c + \to c \times \to a)}{2 Δ}

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C

\frac{(\to a \times \to b + \to b \times \to c + \to c \times \to a)}{3 Δ}

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D

\frac{(\to a \times \to b + \to b \times \to c + \to c \times \to a)}{4 Δ}

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Solution

The correct option is B $\frac{(\to a \times \to b + ¯ ¯ b \times \to c + \to c \times \to a)}{2 Δ}$
$\begin{matrix} V e c t o r p e r p e n d i c u l a r t o t h e p l a n e o f Δ A B C i s - - \to A B \times - - \to A C = (\to b - \to a) \times (\to c - \to a) = \to b \times \to c + \to a \times \to b + \to c \times \to a A r e a o f t r i a n g l e = \frac{1}{2} ∣ ∣ ∣ - - \to A B \times - - \to B C ∣ ∣ ∣ 2 Δ = ∣ ∣ ∣ - - \to A B \times - - \to A C ∣ ∣ ∣ U n i t v e c t o r = \frac{(\to b \times \to c + \to a \times \to b + \to c \times \to a)}{∣ ∣ ∣ \to b \times \to c + \to a \times \to b + \to c \times \to a ∣ ∣ ∣} = \frac{(\to b \times \to c + \to a \times \to b + \to c \times \to a)}{2 Δ} H e n c e, t h e o p t i o n B i s t h e c o r r e c t a n s w e r . \end{matrix}$

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