The altitude through vertex A of triangle ABC | vedclass

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(B) The area of $	riangle ABC$ is given by $	ext{Area} = \frac{1}{2} |\bar{a} 	imes \bar{b} + \bar{b} 	imes \bar{c} + \bar{c} 	imes \bar{a}|$. Al

Vedclass · Accepted Answer

$\frac{|\bar{a} 	imes \bar{b}+\bar{b} 	imes \bar{c}+\bar{c} 	imes \bar{a}|}{|\bar{c}-\bar{b}|}$

Vedclass · Answer

(B) The area of $	riangle ABC$ is given by $	ext{Area} = \frac{1}{2} |\bar{a} 	imes \bar{b} + \bar{b} 	imes \bar{c} + \bar{c} 	imes \bar{a}|$. Also,the area of $	riangle ABC$ is $\frac{1}{2} 	imes 	ext{base} 	imes 	ext{height} = \frac{1}{2} |\bar{c} - \bar{b}| 	imes h$,where $h$ is the altitude from vertex $A$. Equating the two expressions for the area: $\frac{1}{2} |\bar{a} 	imes \bar{b} + \bar{b} 	imes \bar{c} + \bar{c} 	imes \bar{a}| = \frac{1}{2} |\bar{c} - \bar{b}| 	imes h$. Solving for $h$,we get $h = \frac{|\bar{a} 	imes \bar{b} + \bar{b} 	imes \bar{c} + \bar{c} 	imes \bar{a}|}{|\bar{c} - \bar{b}|}$. Thus,the correct option is $B$.

The altitude through vertex $A$ of $\triangle ABC$ with position vectors of points $A, B, C$ as $\bar{a}, \bar{b}, \bar{c}$ respectively is

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