Chorded keyboard (a writing system for the blind) uses either 6 or 8 tactile 'points' from which all letters and numbers are formed. When Louis Braille invented it, it was produced with a needle holing successively all needed points in a cardboard sheet. In 1892, Frank Haven Hall, superintendent of the Illinois Institute for the Education of the Blind, created the Hall Braille Writer, which was like a typewriter with 6 keys, one for each dot in a braille cell. The Perkins Brailler, first manufactured in 1951, uses a 6-key chord keyboard (plus a spacebar) to produce braille output, and has been
Distance from a point to a plane the closest point to the origin may be expressed more succinctly using notation from linear algebra. The expression formula_19 in the definition of a plane is a dot product formula_20, and the expression formula_21 appearing in the solution is the squared norm formula_22. Thus, if formula_23 is a given vector, the plane may be described as the set of vectors formula_24 for which formula_25 and the closest point on this plane is the vector Written in Cartesian form, the closest point is given by formula_27 for formula_28 where and the distance from formula_30 to the hyperplane is Thus in formula_32