Determine whether each equation given represents a line in three dimensions, a plane in three dimensions, or the distance between a point and a plane. Select $$ 4 x+3 y-8 z+6=0 $$ Select $$ d=\frac{|3(7)+(7)(-8)+(4)(5)+9|}{\sqrt{(3)^{2}+(7)^{2}+(4)^{2}}} $$ Select $$ \quad[x, y, z]=[6-6 t,-3+2 t,-3-4 t] $$

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UpStudy AI · Accepted Answer

1. For the equation 
   $$
   4x + 3y - 8z + 6 = 0,
   $$
   this is a linear equation in the variables $$ x $$, $$ y $$, and $$ z $$. In three dimensions, a single linear equation represents a plane.

2. For the equation 
   $$
   d = \frac{|3(7) + 7(-8) + 4(5) + 9|}{\sqrt{(3)^2 + (7)^2 + (4)^2}},
   $$
   this is a formula for the distance $$ d $$ from a given point to a plane. In general, the distance from a point $$(x_0, y_0, z_0)$$ to the plane 
   $$
   Ax + By + Cz + D = 0
   $$
   is given by 
   $$
   d = \frac{|Ax_0 + By_0 + Cz_0 + D|}{\sqrt{A^2 + B^2 + C^2}},
   $$
   so this expression calculates that distance.

3. For the parametric equation 
   $$
   [x, y, z] = [6-6t,\, -3+2t,\, -3-4t],
   $$
   it describes the coordinates $$(x, y, z)$$ in terms of the parameter $$ t $$. A single parameter typically describes a line in three-dimensional space.

Thus:

- $$ 4x + 3y - 8z + 6 = 0 $$ represents a plane.
- $$ d = \frac{|3(7) + 7(-8) + 4(5) + 9|}{\sqrt{(3)^2 + (7)^2 + (4)^2}} $$ is the distance between a point and a plane.
- $$ [x, y, z] = [6-6t, -3+2t, -3-4t] $$ represents a line in three dimensions.
1. The equation $$ 4x + 3y - 8z + 6 = 0 $$ represents a plane. 2. The expression $$ d = \frac{|3(7) + 7(-8) + 4(5) + 9|}{\sqrt{(3)^2 + (7)^2 + (4)^2}} $$ calculates the distance between a point and a plane. 3. The parametric equation $$ [x, y, z] = [6-6t, -3+2t, -3-4t] $$ represents a line in three dimensions.

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