Question

If the non-zero vectors a and b are perpendicular to each other, then the solution of the equation $r\times a=b,$ is given by

• A. $r=xa+\frac{1}{a\cdot a}(a\times b);xϵR$
• B. $r=xb-\frac{1}{b\cdot b}(a\times b);xϵR$
• C. $r=xa\times b,xϵR$
• D. None of these

Answer 1

The correct option is A $r=xa+\frac{1}{a\cdot a}(a\times b);xϵR$

Since $a,b$ and $(a\times b)$ are non-coplanar
$\therefore r=xa+yb+z(a\times b)$ ...(1)
for some scalar $x,y,z$
Now $b=r\times a$
$\therefore b=\{xa+yb+z(a\times b)\}\times a=x\{a\times a\}+y\{b\times a\}+z\{(a\times b)\times a\}=0+y(b\times a)+z\{(a\cdot a)b-(a\cdot b)a\}\because a\cdot b=0\therefore b=y(b\times a)+z(a\cdot a)b$
Comparing coefficients , we get
$y=0$ and $z=\frac{1}{(a\cdot a)}$
Putting the values of $y$ and $z$ in (1), we get
$r=xa+\frac{1}{(a\cdot a)}(a\times b)$