$\overrightarrow r = xa + yb$
Such combination is possible in alternate $(c).$
As $c \times (a \times b) = (c\,.\,b)a - (c\,.\,a)b$ …..$(i)$
Also $(i)$ is perpendicular to $c$
As $c.\,\{ (c\,.\,b)\,a - (c\,.\,a)\,b\} $$ = (c\,.\,a)(c\,.\,b) - (c\,.\,b)(c\,.\,a) = 0$
Thus unit vector perpendicular to $c$ and coplanar with $a,\,b$ is,$\frac{{c \times (a \times b)}}{{|c \times (a \times b)|}}$
Other similar concets :
$(1)$ Unit vector perpendicular to $a$ and coplanar with $b$ and $c$ is $r = \frac{{a \times (b \times c)}}{{|a \times (b \times c)|}}$.
$(2)$ Unit vector perpendicular to $b$ and coplanar with $c$ and $a$ is$r = \frac{{b \times (c \times a)}}{{|b \times (c \times a)|}}$ .
$(b)$ We know ${(a \times b)^2} + {(a\,.\,b)^2} = {a^2}{b^2} = (a\,.\,a)(b\,.\,b)$.
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