Question

If $a,b$ are nonzero vectors and $a$ is perpendicular to $b$, then $a$ has nonzero vector $r$ satisfying $r\cdot a=\alpha ,$ for some scalar $\alpha ,a\times r=b$ is

• A. $\frac{\alpha a+(a+b)}{{\left|a\right|}^2}$
• B. $\frac{\alpha a+a\times b}{{\left|b\right|}^2}$
• C. $\frac{\alpha a-(a\times b)}{{\left|a\right|}^2}$
• D. $\frac{\alpha a-(a\times b)}{{\left|b\right|}^2}$

Answer 1

Answer: (C)

$\frac{\alpha a-(a\times b)}{{\left|a\right|}^2}$

Taking cross product with $a$ on both the sides of $a\times r=b,$
We get $a\times (a\times r)=a\times b$
$\Rightarrow (a\cdot r)a-(a\cdot a)r=a\times b$

$\Rightarrow \alpha a(a\times b)={\left|a\right|}^{2}r$
$\Rightarrow r=\frac{\alpha a-(a\times b)}{{\left|a\right|}^2}$
Clearly this $r$ satisfy $a\times r=b$ and $r\cdot a=\alpha$