ONE MORE STEP TOWARD BETTER TOMORROW : CHAPTER 22 -Worked Out Examples

Friday, 8 August 2014

CHAPTER 22 -Worked Out Examples

Example: 23

Prove that the area of a quadrilateral $ABCD$ can be given by $\dfrac{1}{2}\left| {\overrightarrow {AC} \times \overrightarrow {BD} } \right|$

Solution: 23

The vector area of the quadrilateral can be written as

	$\vec A = \overrightarrow {{\rm{area}}\;\,(\Delta ABC)} + \overrightarrow {{\rm{area}}\;\,(\Delta ACD)}$
	$= \dfrac{1}{2}(\overrightarrow {AB} \times \overrightarrow {AC} ) + \dfrac{1}{2}(\overrightarrow {AC} \times \overrightarrow {AD} )$ To add the areas their directions must be the same
	$= \dfrac{1}{2}(\overrightarrow {AB} \times \overrightarrow {AC} + \overrightarrow {AC} \times \overrightarrow {AD} )$
	$= \dfrac{1}{2}(\overrightarrow {AB} \times \overrightarrow {AC} - \overrightarrow {AD} \times \overrightarrow {AC} )$
	$= \dfrac{1}{2}(\overrightarrow {AB} - \overrightarrow {AD} ) \times \overrightarrow {AC}$
	$= \dfrac{1}{2}\;\overrightarrow {DB} \times \overrightarrow {AC}$

Thus,

$A = \dfrac{1}{2}\left| {\overrightarrow {AC} \times \overrightarrow {BD} } \right|$

Example: 24

Find the perpendicular distance of $C(\vec c)$ from the segment joining $A(\vec a)$ and $B(\vec b)$ , in terms of $\vec a,\;\vec b,\;\vec c$ .

Solution: 24

Now,

	area $(\Delta ABC) = \dfrac{1}{2}\left\| {\overrightarrow {AB} \times \overrightarrow {AC} } \right\|$
	$= \dfrac{1}{2}\left\| {(\vec b - \vec a) \times (\vec c - \vec a)} \right\|$
	$= \dfrac{1}{2}\left\| {(\vec b \times \vec c - \vec b \times \vec a - \vec a \times \vec c)} \right\|$
	$= \dfrac{1}{2}\left\| {(\vec a \times \vec b + \vec b \times \vec c + \vec c \times \vec a)} \right\|$

But area $(\Delta ABC)$ also equals $\dfrac{1}{2} \times AB \times d$

$\Rightarrow \,\,\,\, \dfrac{1}{2} \times AB \times d = \dfrac{1}{2}\left| {\vec a \times \vec b + \vec b \times \vec c + \vec c \times \vec a} \right|$

Since $AB$ equals , we have

$d = \dfrac{{\left| {(\vec a \times \vec b + \vec b \times \vec c + \vec c \times \vec a)} \right|}}{{\left| {\vec a - \vec b} \right|}}$

Friday, 8 August 2014

CHAPTER 22 -Worked Out Examples

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