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Applications of Cross Product

If A, B, C ar...

Question

If A, B, C are three non-collinear points with position vectors $\to a, \to b, \to c$ , respectively, then show that the length of the perpendicular from C on AB is $\frac{| (\to a \times \to b) + (\to b \times \to c) + (\to c \times \to a) |}{| \to b - \to a |}$ .

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Solution

Consider $△ A B C$ , with base AB.
So, length of the perpendicular from C on AB is the height of the triangle, $h$ .
Method 1:
$h = A C sin θ$ , where $θ$ is the angle between AB and AC.
$\Rightarrow h = \frac{| \to A B | | \to A C | sin θ}{| \to A B |} = \frac{| \to A B \times \to A C |}{| \to A B |} = \frac{| (\to b - \to a) \times (\to c - \to a) |}{| \to b - \to a |} = \frac{| (\to b \times \to c) - (\to a \times \to c) - (\to b \times \to a) |}{| \to b - \to a |}$
$\Rightarrow h = \frac{| (\to a \times \to b) + (\to b \times \to c) + (\to c \times \to a) |}{| \to b - \to a |}$
Method 2:
$h = \frac{A r e a o f T r i a n g l e}{\frac{1}{2} A B} = \frac{\frac{1}{2} | \to A B \times \to A C |}{\frac{1}{2} | \to A B |}$
$= \frac{| \to A B \times \to A C |}{| \to A B |} = \frac{| (\to b - \to a) \times (\to c - \to a) |}{| \to b - \to a |} = \frac{| (\to b \times \to c) - (\to a \times \to c) - (\to b \times \to a) |}{| \to b - \to a |}$
$\Rightarrow h = \frac{| (\to a \times \to b) + (\to b \times \to c) + (\to c \times \to a) |}{| \to b - \to a |}$
Hence, proved.

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