Let a→=i^+j^;b→=2i^-k^. Then vector r→satisffing the equations r→×a→=b→×a→ and r→×b→=a→×b→ is

# Let $\stackrel{\to }{a}=\stackrel{^}{i}+\stackrel{^}{j};\stackrel{\to }{b}=2\stackrel{^}{i}-\stackrel{^}{k}$. Then vector $\stackrel{\to }{r}$satisffing the equations $\stackrel{\to }{r}×\stackrel{\to }{a}=\stackrel{\to }{b}×\stackrel{\to }{a}$ and $\stackrel{\to }{r}×\stackrel{\to }{b}=\stackrel{\to }{a}×\stackrel{\to }{b}$ is

1. A

$\stackrel{^}{i}-\stackrel{^}{j}+\stackrel{^}{k}$

2. B

$3\stackrel{^}{i}-\stackrel{^}{j}+\stackrel{^}{k}$

3. C

$3\stackrel{^}{i}+\stackrel{^}{j}-\stackrel{^}{k}$

4. D

$\stackrel{^}{i}-\stackrel{^}{j}-\stackrel{^}{k}$

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### Solution:

$\stackrel{\to }{r}×\stackrel{\to }{a}=\stackrel{\to }{b}×\stackrel{\to }{a}\text{\hspace{0.17em}\hspace{0.17em}}or\text{\hspace{0.17em}}\left(\stackrel{\to }{r}-\stackrel{\to }{b}\right)×\stackrel{\to }{a}=0\text{\hspace{0.17em}}$

$\stackrel{\to }{r}×\stackrel{\to }{b}=\stackrel{\to }{a}×\stackrel{\to }{b}\text{\hspace{0.17em}\hspace{0.17em}}or\text{\hspace{0.17em}}\left(\stackrel{\to }{r}-\stackrel{\to }{a}\right)×\stackrel{\to }{b}=0\text{\hspace{0.17em}}$

If $\stackrel{\to }{r}=x\stackrel{^}{i}+y\stackrel{^}{j}+z\stackrel{^}{k}$, then

$\left|\begin{array}{ccc}\stackrel{^}{i}& \stackrel{^}{j}& \stackrel{^}{k}\\ x-2& y& z+1\\ 1& 1& 0\end{array}\right|=0\text{\hspace{0.17em}}and\text{\hspace{0.17em}}\left|\begin{array}{ccc}\stackrel{^}{i}& \stackrel{^}{j}& \stackrel{^}{k}\\ x-1& y-1& z\\ 2& 0& -1\end{array}\right|=0$

$\begin{array}{l}⇒z+1=0,x-y=2\\ and\text{\hspace{0.17em}\hspace{0.17em}}y-1=0,\text{\hspace{0.17em}}x-1+2z=0\\ ⇒x=3,y=1,z=-1\end{array}$  