At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.
I read somewhere that it is the distance to the origin. Is this correct? Also distance to what? The center of the plane?
Do you know about vectors and the dot product between 2 vectors ?
yes this is what I am learning. not quite sure I have a full grasp on it just yet
Let me try again. let u be a vector with length 1, |u|= 1 \[ v \cdot u = |v| |u| \cos \theta = |v| \cos \theta \] at θ = 0 you get |v| cos 0 = |v| |dw:1371566156694:dw|
let's say the length of vector v in the above picture is K now consider another vector w, where \[ w \cdot u = K \] |dw:1371566296257:dw|
notice that the "head" of w "points" to a specific spot. If we look at all w vectors where u dot w = K, the "spots" or points described by the "head" of all vectors w will describe a line in 2D or a plane in 3D
maybe this will help http://tutorial.math.lamar.edu/Classes/CalcII/EqnsOfPlanes.aspx
So in something like x +2y + 3z = 5, what does the 5 tell me?
the coefficients tell me the information for the normal vector to the plane, but i was just curious what the 5 means?
the normal vector to the plane is (1,2,3) its magnitude is sqrt(1+4+9)= sqrt(14) let P= (x,y,z) so that (1,2,3) dot P =5 means 1*x + 2*y + 3*z = 5 i.e. the equation of a plane if we normalize the normal vector by its magnitude (1,2,3)/sqrt(14) dot P = 5/sqrt(14) P dot with a unit vector gives the "projection" of P onto the normal It is the length of P in the direction of the normal 5/sqrt(14) tells you the distance from the origin (0,0,0) to the nearest point on the plane from the origin is 5/sqrt(14)
so the 5 by itself does not tell you much. You must divide it by the length of the normal, and that tells you the distance between the origin and the plane (defined to be the shortest distance between the plane and the origin)