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 $b=xa+y(a\times c)(1)$
$\therefore \beta =a\cdot b=a\cdot [xa+y(a\times c)]$
$=x|a{|}^2+ya\cdot (a\times c)=x{a}^2$
$\Rightarrow x=\beta /a^2$
And, $c=a\times b=a\times [xa+y(a\times c)]$
$=xa\times a+ya\times (a\times c)$
$=0+y(a\cdot c)a-y(a\cdot a)c$
$=y\{a\cdot (a\times b)\}a-y{a}^{2}c=-y{a}^{2}c$
$\Rightarrow y=-1/a^2$
$\therefore$ from $(1),b=(\beta a-a\times c)/a^2.$