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The equation $d=\frac{|Ax+By+Cz+D|}{\left(\sqrt{{A}^{2}+{B}^{2}+{C}^{2}}\right)}$ gives is the distance between a plane and a point as the value 'd'. What is the unit of this value? Why is the distance between the plane and the point not a vector?
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Haleigh Vega
The equation $d=\frac{|Ax+By+Cz+D|}{\sqrt{{A}^{2}+{B}^{2}+{C}^{2}}}$ gives us the distance a point from a plane in ${\mathbb{R}}^{3}$. Here we discuss about Euclidean space and distances measured by physical units, here obey on $x$, $y$ and $z$ and $A$, $B$, $C$ and $D$ are constant scalars belong to ${\mathbb{R}}^{3}$ without units. Then the unit of d is the same as variables have.
Also, the distance is not a vector, it is the smallest way between the point and a point on plane measured without direction.
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Briana Petty
Distances have units of length. You can see this from the formula, where the numerator has units like $Ax$ and the denominator has units like $A$, so the ratio has units of $x$. You can certainly define a vector from the point to the nearest point on the plane, but that is not this formula. This formula gives the magnitude of that vector.