Name: Distance Between Two Parallel Lines in 3D
Uploaded: 2019-07-19
Duration: 6 min 41 s
Description: What do you know about parallel lines in three-dimensional space? Do they stay at a constant distance apart? How do we find that distance? This video answers...

Question

Derive the vector equation of a plane in the normal form and hence obtain the Cartesian equation of the plane.

Accepted Answer

Consider a plane whose perpendicular distance from the origin is $d$ , where $d \neq 0$ .

If $\vec{O N} = d \hat{n}$

Let $p$ be any point on the plane. Thus,

$\vec{N P} \cdot \vec{O P} = 0 . . . (1)$

But $\vec{N P} = \vec{O N} - \vec{O N} = \vec{r} - d \hat{n}$

and $\vec{O N} = d \hat{n}$

$∴ (1) \Rightarrow (\vec{r} - d \hat{n}) \cdot d \hat{n}$

$\Rightarrow (\vec{r} - d \hat{n}) \cdot \hat{n} = 0 (∵ d \neq 0)$

$\Rightarrow \vec{r} \cdot \hat{n} - d (\hat{n} \cdot \hat{n}) = 0$

$\Rightarrow \vec{r} \cdot \hat{n} - d (1) = 0 [∵ \hat{n} \cdot \hat{n} = |\hat{n}| = 1]$

$\Rightarrow \vec{r} \cdot \hat{n} = d . . . (2)$

Cartesian form:

Let $\vec{r} = x \hat{i} + y \hat{j} + z \hat{k}$

and $\hat{n} = l \hat{i} + m \hat{j} + n \bar{k}$

Thus, $(2) \Rightarrow (x \hat{i} + y \hat{j} + z \hat{k}) \cdot (l \hat{i} + m \hat{j} + n \bar{k}) = d$

$\Rightarrow l x + m y + n z = d$

This is the required equation of plane in normal form.

Derive the vector equation of a plane in the normal form and hence obtain the Cartesian equation of the plane.

Important Questions on Three Dimensional Geometry

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