# Math Help - Question on 3-d vectors

1. ## Question on 3-d vectors

A triangle has vertices at the points A(5,0,0), B(0,5,0), and C(0,0,5). Identify the point in the interior of the triangle that is closest to the origin.

2. Originally Posted by dalbir4444
A triangle has vertices at the points A(5,0,0), B(0,5,0), and C(0,0,5). Identify the point in the interior of the triangle that is closest to the origin.
1. The points A, B, C define a plane:

$px,y,z)=(5,0,0)+r(5,-5,0+s(5,0,-5)" alt="px,y,z)=(5,0,0)+r(5,-5,0+s(5,0,-5)" />

2. p in normal form:

$\vec n = (5,-5,0) \times (5,0,-5) = (5,5,5)$ Thus

$p: x+y+z=5$

3. The point F lies on the line through the origin perpendicular to p:

$(x,y,z)=r (1,1,1)$

4. Calculate the point of intersection:

$r+r+r = 5~\implies~r=\dfrac53$

Therefore $F\left(\dfrac53\ ,\ \dfrac53\ ,\ \dfrac53\right)$