Let a plane P pass through the point

Question:

Let a plane P pass through the point $(3,7,-7)$ and

contain the line, $\frac{x-2}{-3}=\frac{y-3}{2}=\frac{z+2}{1}$. If distance

of the plane $P$ from the origin is $d$, then $d^{2}$ is equal to

 

 

Solution:

$\overrightarrow{\mathrm{BA}}=(\hat{\mathrm{i}}+4 \hat{\mathrm{j}}-5 \hat{\mathrm{k}})$

$\overrightarrow{\mathrm{BA}}=(\hat{\mathrm{i}}+4 \hat{\mathrm{j}}-5 \hat{\mathrm{k}})$

$\overrightarrow{\mathrm{BA}} \times \vec{\ell}=\overrightarrow{\mathrm{n}}=\left|\begin{array}{ccc}\hat{\mathrm{i}} & \hat{\mathrm{j}} & \hat{\mathrm{k}} \\ -3 & 2 & 1 \\ 1 & 4 & -5\end{array}\right|$

$a \hat{i}+b \hat{j}+c \hat{k}=-14 \hat{i}-\hat{j}(14)+\hat{k}(-14)$

$\mathrm{a}=1, \mathrm{~b}=1, \mathrm{c}=1$

Plane is $(x-2)+(y-3)+(z+2)=0$

$x+y+z-3=0$

$\mathrm{d}=\sqrt{3} \Rightarrow \mathrm{d}^{2}=3$

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