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Ratios of Distances between Centroid, Circumcenter, Incenter and Orthocenter of Triangle

If a⃗, b⃗ and...

Question

If $\to a, \to b a n d \to c$ determine the vertices of a triangle, show that $\frac{1}{2} [\to b \times \to c + \to c \times \to a + \to a \times \to b]$ gives the vector area of the triangle. Hence, deduce the condition that the three points $\to a, \to b a n d \to c$ are collinear. Also, find the vector normal to the plane of the triangle.

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Solution

Since, $\to a, \to b a n d \to c$ are the vertices of a $△$ ABC as shown.

$Area of △ A B C = \frac{1}{2} | \to A B \times \to A C | N o w, \to A B = \to b - \to a and \to A C = \to c - \to a ∴ Area of △ A B C = \frac{1}{2} | \to b - \to a \times \to c - \to a | = \frac{1}{2} | \to b \times \to c - \to b \times \to a - \to a \times \to c + \to a \times \to a | = \frac{1}{2} | \to b \times \to c + \to a \times \to b + \to c \times \to a + \to 0 | = \frac{1}{2} | \to b \times \to c + \to a \times \to b + \to c \times \to a | . . . (i)$

For three points to be collinear, area of the $△$ ABC should be equal to zero.

$\Rightarrow \frac{1}{2} [\to b \times \to c + \to c \times \to a + \to a \times \to b] = 0 \Rightarrow \to b \times \to c + \to c \times \to a + \to a \times \to b = 0 (i i)$

This is the required condition for collinearity of three points $\to a, \to b a n d \to c$

Let $^n$ be the unit vector normal to the plane of the $△$ ABC.

$∴^n = \frac{\to A B \times \to A C}{| \to A B \times \to A C |} = \frac{\to a \times \to b + \to b \times \to c + \to c \times \to a}{| \to a \times \to b + \to b \times \to c + \to a \times \to a |}$

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Ratios of Distances between Centroid, Circumcenter, Incenter and Orthocenter of Triangle

Standard XII Mathematics

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