Answer
(a) $d=\dfrac {|a \cdot (b \times c)|} {| a \times b|}$
(b) $\frac{17}{7}$
Work Step by Step
(a) Consider a parallelepiped defined by vectors $a,b, c$
$V=AH=|c \cdot (a \times b)|$
Area of a parallelgram is defined as $| a \times b|$.
Thus, $d= \frac{V}{A}= \frac {|c \cdot (a \times b)|} {| a \times b|}$
Because $d \perp a$ and $b$ this means that $d=H$
By the property of cross product : $c \cdot (a \times b)=a \cdot (b \times c)$
Hence, $d=\dfrac {|a \cdot (b \times c)|} {| a \times b|}$
(b) $a=QR= \lt 0-1, 0-2,0-0= \lt -1,2,0 \gt$
$b=QS=\lt 0-1, 0-0,3-0= \lt -1,0,3 \gt$
$c=QP=\lt 2-1, 1-0,4-0= \lt 1,1,4 \gt$
From part (a), we have
$d=\dfrac {|a \cdot (b \times c)|} {| a \times b|}=\dfrac{|\lt -1,2,0 \gt \cdot ( \lt -1,0,3 \gt \times \lt 1,1,4 \gt)|}{|\lt -1,2,0 \times \lt -1,0,3 \gt|}$
$=\dfrac{|\lt -1,2,0 \gt \cdot ( \lt -1,0,3 \gt \times \lt 1,1,4 \gt)|}{|\lt -1,2,0 \times \lt -1,0,3 \gt|}$
$=\dfrac{|\lt -1,2,0 \gt \cdot \lt -3,7,-1 \gt|}{|\lt 6,3,2 \gt|}$
$=\frac{|17|}{\sqrt {49}}$
$=\frac{17}{7}$
