Question

The equation of the plane through $(1,2,-3)$ and $(2,-2,1)$ and parallel to X-axis is:

• A. $y-z+1=0$
• B. $y-z-1=0$
• C. $y+z-1=0$
• D. $y+z+1=0$

Answer 1

The correct option is D $y+z+1=0$

The points $(1,2,-3)$ and $(2,-2,1)$

$\overrightarrow{u}=\left(\begin{array}{c}2\\ -2\\ 1\end{array}\right)=\left(\begin{array}{c}1\\ 2\\ -3\end{array}\right)=\left(\begin{array}{c}1\\ -4\\ 4\end{array}\right)$

A vector direction for the x-axis is :

$\overrightarrow{v}=\left(\begin{array}{c}1\\ 0\\ 0\end{array}\right)$

A normal vector to the plane is :

$\overrightarrow{n}=\overrightarrow{u}\times \overrightarrow{v}$

$=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& -4& 4\\ 1& 0& 0\end{array}\right|$

$=\hat{i}(0-0)-\hat{j}(0-4)+\hat{k}(0+4)$

$=<0,4,4>$

$\Rightarrow 0(x-1)+4(y-2)+4(z+3)=0$

$\Rightarrow 4y-8+4z+12=0$

$\Rightarrow 4y+4z+4=0$

$\Rightarrow y+z+1=0$