Question

If P( 2,3,4) is the foot of the perpendicular from origin to a plane , then write the vector equation of this plane.

Answer 1

Answer: The  vector equation of the plane is,

Step-by-step explanation:

P( 2,3,4) is the foot of the perpendicular from origin to a plane,

Since, The normal vector of the plane,

OP = $\overrightarrow{n}$ = 2i+3j+4k

The distance of the plane from the point,

$d=|OP|=\sqrt{2^2+3^2+4^2}=\sqrt{29}$

Now, The unit normal vector,

$\hat{n}=\frac{\overrightarrow{n}}{|n|}$

$=\frac{2i+3j+4k}{\sqrt{29}}$

Since, the vector equation of a plan is,

$\overrightarrow{r}.\hat{n}=d$

$\implies \overrightarrow{r}.\frac{\overrightarrow{n}}{|n|} = d$

$\implies \overrightarrow{r}. \frac{2i+3j+4k}{\sqrt{29}} = \sqrt{29}$

$\implies \overrightarrow{r}. ( 2i+3j+4k)=29$

Where,

$\overrightarrow{r}=xi+yj+zk$