Multivariable calculus equation of a plane

[iScience]

equation of the form

ax+bx+cx=a₀+b₀+c₀

ax+bx+cx=d


what exactly does "d" represent? is it the distance of a point from the plane? or is it the shifting of the plane along the normal vector of the plane?
and how does "d" represent that again?

 

[MisterX]

A way to write an equation for a plane in $\mathbb{R}^3$ is
$$\vec{w} \cdot \vec{r} = d$$and the plane contains the point $$d\frac{ \vec{w}}{\left\|\vec{w}\right\|^2}$$ You can check that the above is true. Now if $\vec{u}$ and $\vec{v}$ are parallel to the plane and one is not a scalar multiple of the other, then we could write the plane as $$\vec{r}(s,t) = d\frac{ \vec{w}}{\left\|\vec{w}\right\|^2} + s\vec{u} + t\vec{v}$$ since we can always parameterize the plane as a point on the plane plus scalar multiples of two spanning vectors added on.

So you could think that the plane has been shifted by $d\frac{ \vec{w}}{\left\|\vec{w}\right\|^2}$, which is along the normal. It is somewhat ambiguous perhaps because some other shifts in other directions would result in the same plane.

The above applies to planes in $\mathbb{R}^3$.

 

[LCKurtz]

equation of the form

ax+bx+cx=a₀+b₀+c₀

ax+bx+cx=d


what exactly does "d" represent? is it the distance of a point from the plane? or is it the shifting of the plane along the normal vector of the plane?
and how does "d" represent that again?

I guess you mean $ax + by + cz = d$, not what you wrote. In that case the quantity$$\frac{|d|}{\sqrt{a^2+b^2+c^2}}$$gives the distance of the plane from the origin.

 

[HallsofIvy]

Also, if x= y= 0 then cz= d so z= d/c, if x= z= 0 then by= d so y= d/b, and if y= z= 0 then ax= d so x= d/a. That is, (d/a, 0, 0), (0, d/b, 0), and (0, 0, d/c) are the x, y, and z intercepts of the plane.

 

[iScience]

I guess you mean $ax + by + cz = d$, not what you wrote. In that case the quantity$$\frac{|d|}{\sqrt{a^2+b^2+c^2}}$$gives the distance of the plane from the origin.

yes that's what i meant, could you show me how this yields the distance of the plane from the origin?
what does d itself represent?

 

[LCKurtz]

I guess you mean $ax + by + cz = d$, not what you wrote. In that case the quantity$$\frac{|d|}{\sqrt{a^2+b^2+c^2}}$$gives the distance of the plane from the origin.

yes that's what i meant, could you show me how this yields the distance of the plane from the origin?
what does d itself represent?

Look in your calculus book how to calculate the distance from a plane to a point and use it to calculate the distance to (0,0,0).

$d$ by itself doesn't have any particular meaning. If you multiply the equation through by $2$ you will have the same plane but now $2d$ on the right side. So $d$ in relation to the coefficients is what is significant. And the formula above gives geometric meaning to it.