Discussion Forum

The vector(s) which is/are coplanar with vectors   $\widehat{i}+\widehat{j}+2\widehat{k}$ and $\widehat{i}+2\widehat{j}+\widehat{k}$ , are  perpendicular to the vector  $\widehat{i}+\widehat{j}+\widehat{k}$ is/are 

Answer:

Explanation:

Let $a= \widehat{i}+\widehat{j}+2\widehat{k},b=\widehat{i}+2\widehat{j}+\widehat{k}$ and

$c= \widehat{i}+\widehat{j}+\widehat{k}$

$\therefore$ Avector coplanar to a and b and perpendicular to c

 Now  $\lambda ( a \times b) \times c$

 $\Rightarrow$         $\lambda {(a.c)b-(b.c)a}$

 $\Rightarrow$  $\lambda {(1+1+4)(\widehat{i}+2 \widehat{j}+\widehat{k})}$

                                           ${-(1+2+1)(\widehat{i}+\widehat{j}+2 \widehat{k}})$

$\Rightarrow$  $\lambda (6 \widehat{i}+12 \widehat{j}+6 \widehat{k}-6\widehat{i}-6\widehat{j}-12\widehat{k})$

 $\Rightarrow$  $\lambda(6 \widehat{j}-6 \widehat{k}) \Rightarrow  6 \lambda (\widehat{j}-\widehat{k})$

 for  $\lambda=\frac{1}{6} \Rightarrow$ Option (a) is correct

for $\lambda=-\frac{1}{6} \Rightarrow$ Option (d) is correct