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
Dot product identity, also known as "Lagrange's formula" may be remembered as "BAC minus CAB", keeping in mind which vectors are dotted together. This formula finds application in simplifying vector calculations in physics. In physics, vector magnitude is a scalar in the physical sense, i.e. a physical quantity independent of the coordinate system, expressed as the product of a numerical value and a physical unit, not just a number. The dot product is also a scalar in this sense, given by the formula, independent of the coordinate system. Examples include: For vectors with complex entries, using the given definition of the dot