The position vectors of the vertices of a qua | vedclass

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(C) Let $a, b, c$ and $d$ be the position vectors of the vertices $A, B, C$ and $D$ of a quadrilateral respectively.Let $E, F, G$ and $H$ be the midpo

Vedclass · Accepted Answer

$\frac{1}{4}\,\left| a	imes b+b	imes c+c	imes d+d	imes a ight|$

Vedclass · Answer

(C) Let $a, b, c$ and $d$ be the position vectors of the vertices $A, B, C$ and $D$ of a quadrilateral respectively.Let $E, F, G$ and $H$ be the midpoints of the sides $AB, BC, CD$ and $DA$ respectively.The position vectors of these midpoints are:$\overrightarrow{OE} = \frac{a+b}{2}, \overrightarrow{OF} = \frac{b+c}{2}, \overrightarrow{OG} = \frac{c+d}{2}, \overrightarrow{OH} = \frac{d+a}{2}$Now,the sides of the quadrilateral $EFGH$ are:$\overrightarrow{EF} = \overrightarrow{OF} - \overrightarrow{OE} = \frac{b+c}{2} - \frac{a+b}{2} = \frac{c-a}{2}$$\overrightarrow{FG} = \overrightarrow{OG} - \overrightarrow{OF} = \frac{c+d}{2} - \frac{b+c}{2} = \frac{d-b}{2}$Since $EFGH$ is a parallelogram (Varignon's Theorem),its area is given by the magnitude of the cross product of two adjacent sides:$	ext{Area} = |\overrightarrow{EF} 	imes \overrightarrow{FG}| = |\frac{c-a}{2} 	imes \frac{d-b}{2}|$$= \frac{1}{4} |(c-a) 	imes (d-b)|$$= \frac{1}{4} |c 	imes d - c 	imes b - a 	imes d + a 	imes b|$$= \frac{1}{4} |a 	imes b + b 	imes c + c 	imes d + d 	imes a|$

List-$I$	List-$II$
$(A)$ $[\mathbf{a} \mathbf{b} \mathbf{c}]$	$1. \|\mathbf{a}\|\|\mathbf{b}\|\cos(\mathbf{a}, \mathbf{b})$
$(B)$ $(\mathbf{c} \times \mathbf{a}) \times \mathbf{b}$	$2. (\mathbf{a} \cdot \mathbf{c})\mathbf{b} - (\mathbf{a} \cdot \mathbf{b})\mathbf{c}$
$(C)$ $\mathbf{a} \times (\mathbf{b} \times \mathbf{c})$	$3. \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})$
$(D)$ $\mathbf{a} \cdot \mathbf{b}$	$4. \|\mathbf{a}\|\|\mathbf{b}\|$
	$5. (\mathbf{b} \cdot \mathbf{c})\mathbf{a} - (\mathbf{a} \cdot \mathbf{b})\mathbf{c}$

The position vectors of the vertices of a quadrilateral $ABCD$ are $a, b, c$ and $d$ respectively. The area of the quadrilateral formed by joining the midpoints of its sides is

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