Question

Consider the plane $(x,y,z)=(0,1,1)+\lambda (1,-1,1)+\mu (2,-1,0).$The distance of this plane from the origin is

• A. $1/3$
• B. $\sqrt{3}/2$
• C. $\sqrt{3/2}$
• D. $2\sqrt{3}$

Answer 1

The correct option is C $\sqrt{3/2}$

Given $(x,y,z)=(0,1,1)+\lambda (1,-1,1)+\mu (2,-1,0)$
$\Rightarrow xi+yj+zk=j+k+\lambda (i-j+k)+\mu (2i-j)$
comparing coefficients of $i,j$ and $k$
$\lambda +2\mu =x..\left(1\right)$
$-\lambda -\mu =y-1..\left(2\right)$
$-2\mu =y+z-2..\left(3\right)$
eliminating $\lambda$ and $\mu$ from (1), (2) and (3) we get
$x+2y+z+3=0$
Hence distance of this plane from origin is
$=\left|\frac{3}{\sqrt{1^2+2^2+1^2}}\right|=\sqrt{3/2}$