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Multiplication of a Vector by a Scalar

If a,b,c are ...

Question

If $\to a, \to b, \to c$ are position vectors of the vertices of a triangle $A B C$ respectively, then length of the perpendicular drawn from $C$ to $A B$ is

\frac{∣ ∣ ∣ \to a \times \to b + \to b \times \to c + \to c \times \to a ∣ ∣ ∣}{| \to a - \to b |}

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\frac{∣ ∣ ∣ \to a \times \to b + \to b \times \to c + \to c \times \to a ∣ ∣ ∣}{| \to b - \to c |}

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\frac{∣ ∣ ∣ \to a \times \to b + \to b \times \to c + \to c \times \to a ∣ ∣ ∣}{| \to a - \to c |}

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∣ ∣ ∣ \to a \times \to b + \to b \times \to c + \to c \times \to a ∣ ∣ ∣

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Solution

The correct option is A $\frac{∣ ∣ ∣ \to a \times \to b + \to b \times \to c + \to c \times \to a ∣ ∣ ∣}{| \to a - \to b |}$

Let $A B C$ is a given triangle having position vector of vertices be $\to a, \to b and \to c$ respectively
Area of triangle $= \frac{1}{2} (A B) (C P)$ $= \frac{1}{2} (A B) (A C) sin A$ $\Rightarrow - - \to A C \times - - \to A B = ∣ ∣ ∣ \to a \times \to b + \to b \times \to c + \to c \times \to a ∣ ∣ ∣$ $\Rightarrow | - - \to A B | | - - \to A C | sin A = ∣ ∣ ∣ \to a \times \to b + \to b \times \to c + \to c \times \to a ∣ ∣ ∣$ $\Rightarrow | \to b - \to a | | - - \to C P | = ∣ ∣ ∣ \to a \times \to b + \to b \times \to c + \to c \times \to a ∣ ∣ ∣$ $\Rightarrow | - - \to C P | = | - - \to A C | sin A = \frac{∣ ∣ ∣ \to a \times \to b + \to b \times \to c + \to c \times \to a ∣ ∣ ∣}{| \to b - \to a |}$ $\Rightarrow | - - \to C P | = \frac{∣ ∣ ∣ \to a \times \to b + \to b \times \to c + \to c \times \to a ∣ ∣ ∣}{| \to a - \to b |}$

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