so if you have a plane with equation r.n = k, apparently you can find the distance from the plane to the origin by doing k / |n|, why/how does this work?
#Distance from plane to origin.
digital yoke
bleak flameBOT
wooden junco
First recall that, geometrically, the dot product of two vectors, u and v, is defined as the product of the magnitudes of u and v and the cosine of the angle between them:
u • v = |u| |v| cos(theta)
Now consider a plane p with normal n.
Since n remains the same everywhere on p, we know the closest point, lets call this d, from the origin to p will be found somewhere along the trajectory of n. It follows that the angle between d and n will subsequently be 0. Therefore, we can express p as the dot product between d and n:
d • n = k
Equivalently, we have
|d| |n| cos(0) = k
=> |d| |n| = k