Question

If $a \cdot b=\beta$ and $a \times b=c$, then $\beta$ is equal to

• A. $(\beta a-a \times c) / a^{2}$
• B. $(\beta a+a \times c) / a^{2}$
• C. $(\beta c-a \times c) / a^{2}$
• D. $(\beta c+a \times c) / a^{2}$

Answer 1

Answer: (A)

Here, $a$ and $c=a \times b$ are non-collinear vectors.
$\therefore $ Let $ b=x a+y(a \times c)\,\,\,\, (1)$
$\therefore \beta=a \cdot b=a \cdot[x a+y(a \times c)]$
$=x|a|^{2}+y a \cdot(a \times c)=x a^{2} $
$\Rightarrow x=\beta / a^{2}$
And, $c=a \times b=a \times[x a+y(a \times c)]$
$=x a \times a+y a \times(a \times c) $
$=0+y(a \cdot c) a-y(a \cdot a) c$
$=y\{a \cdot(a \times b)\} a-y a^{2} c=-y a^{2} c $
$\Rightarrow y=-1 / a^{2}$
$\therefore $ from $(1), b=(\beta a-a \times c) / a^{2} .$