In $\triangle A B C$ if $A(\alpha), B(\beta)$ and $C(\gamma)$ are the position vectors of the vertices, then…

1. $|\alpha \times \beta|+|\beta \times \gamma|+|\gamma \times \alpha|$
2. , $|\alpha \times \beta+\beta \times \gamma+\gamma \times \alpha|$
3. $\frac{|\alpha \times \beta+\beta \times \gamma+\gamma \times \alpha|}{|\alpha-\beta|}$
4. $\frac{|\alpha \times \beta+\beta \times \gamma+\gamma \times \alpha|}{|\gamma-\beta|}$

Solution

Let $A B C$ be a triangle and $\alpha, \beta, \gamma$ be the position vector of the vertices $A, B, C$ respectively. Let $A M$ be the perpendicular from $A$ to $B C$ Then, Area of $\triangle A B C$
$ \begin{aligned} & =\frac{1}{2}(B C) \cdot(A M) \\ & =\frac{1}{2}|\mathbf{B C}||\mathbf{A M}| \end{aligned} $ Area of $\triangle A B C$ $ =\frac{1}{2}|\alpha \times \beta+\beta+\gamma+\gamma \times \alpha| $ $ \begin{array}{lrl} \Rightarrow & \frac{1}{2}|\mathbf{B C}||\mathbf{A M}|=\frac{1}{2}|\alpha \times \beta+\beta \times \gamma+\gamma \times \alpha| \\ \Rightarrow & |\mathbf{A M}|=\frac{|\alpha \times \beta+\beta \times \gamma+\gamma \times \alpha|}{|\gamma-\beta|} \end{array} $

Asked in: AP EAMCET 2018 (24 Apr Shift 1)

Practice more Vectors questions on Aicharya