# Thread: closest point

1. ## closest point

I need to find the point in the plane x + 2y + 3z = 6 that is nearest the origin.

2. Originally Posted by chaddy
I need to find the point in the plane x + 2y + 3z = 6 that is nearest the origin.
To do this, find the normal vector to the plane that passes through the origin.

3. How would I find the normal vector?

4. Does anyone know?

5. Originally Posted by chaddy
How would I find the normal vector?
For the plane ax + by + cz = d, a normal vector is ai + bj + ck.

6. Originally Posted by mr fantastic
For the plane ax + by + cz = d, a normal vector is ai + bj + ck.
So the parametric equations of the line passing through the origin and in the direction of this normal vector will be:

x = at
y = bt
z = ct

The point where this line cuts the plane will be the point that's closest to the origin:

$\displaystyle a(at) + b(bt) + c(ct) = d \Rightarrow t = \frac{d}{a^2 + b^2 + c^2}$.

Therefore the coordinates of the point in the plane closest to the origin are:

$\displaystyle x = at = \frac{ad}{a^2 + b^2 + c^2}$.

$\displaystyle y = bt = \frac{bd}{a^2 + b^2 + c^2}$.

$\displaystyle x = at = \frac{cd}{a^2 + b^2 + c^2}$.

And as a bonus, the answer to what the closest distance is equal to is ..........

7. $\displaystyle x = \frac{6}{14}$

$\displaystyle y = \frac{12}{ 14}$

$\displaystyle z = \frac{18}{ 14}$

So the point closest to the origin would be < $\displaystyle \frac{3}{7}$, $\displaystyle \frac{6}{7}$, $\displaystyle \frac{9}{7}$ >

8. Originally Posted by chaddy
$\displaystyle x = \frac{6}{14}$

$\displaystyle y = \frac{12}{ 14}$

$\displaystyle z = \frac{18}{ 14}$

So the point closest to the origin would be < $\displaystyle \frac{3}{7}$, $\displaystyle \frac{6}{7}$, $\displaystyle \frac{9}{7}$ >