Find the distance of the point $( 1,2,3 )$ to ,,,,,,,,,,,,,,

Question

Find the distance of the point $( 1,2,3 )$ of the plane
$3x-2y+5z+17=0$ .

Answer 1

Hint: from analytic formulas you know that the distance beteween a plane and a point is $$\frac{ax_0 + by_o+cz_o+d}{\sqrt{a^2+b^2+c^2}}$$ given a plane of equation $ax+by+cz+d=0$, like in this problem...

So substituting we get $$ d = \frac{3\cdot 1 -2\cdot 2 +5\cdot 3 +17}{\sqrt{3^2+(-2)^2+5^2}} = \frac{3-4+15+17}{\sqrt{9+4+25}}=\frac{31}{\sqrt{38}}$$

Answer 2

Hint: First, can you find the normal vector $\vec n$to the plane from the equation? Now parameterize a line through $(1,2,3)$ and in the direction of the normal as $(1,2,3)+t\vec n$. Find the parameter that represents the point on the line that is in the plane.