Question

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

• A. $r=xa+\frac{1}{a.a}(a\times b)$
• B. $r=xb+\frac{1}{b.b}(a\times b)$
• C. $r=x(a\times b)$
• D. none of these

Answer 1

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

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 $b=r\times a$

$\therefore b=\{xa+yb+z(a\times b)\}\times a$

$=x(a\times a)+y(b\times a)+z\{(a\times b)\times a\}$

$=0+y(b\times a)+z\{(a.a)b-(a.b)a\}$

Comparing the coefficients, we get $y=o$ and $z=\frac{1}{(a.a)}$.

Putting the values of $y$ and $z$ in (1), we get

$r=xa+\frac{1}{a.a}(a\times b)$