Scalar product.tif

94 Formula for the scalar product in Cartesian coordinates

This model shows two vectors (green and lavender) resolved into their components parallel to the coordinate axes (in red, blue, and yellow). The components of the green vector are projected onto the lavender vector by means of the white perpendiculars. These divide the lavender vector into three pieces, making visible the fact that the projection of the green vector is the sum of the projections of its components, a key fact in proving the formula for the scalar product in terms of Cartesian coordinates. This is probably model 230 in Baker’s catalogue, titled cos u = cosa cosa' + cosb cosb' + cosg cosg' . Here u is the angle between the green and lavender vectors, a, b, and g are the direction cosines of the lavender vector (angles with the coordinate axes), and a', b', and g' are the direction cosines of the green vector.