Dimensions and Identity of Vector Direction.

Extract

In an equation with real coefficients, if the terms are vectors, they must be codirectional if the equation is possible with all the quantities real. Thus to obtain a geometrical solution of the equation a cos θ + b sin θ =c, a and b must be drawn at right angles to each other since a cos θ makes an angle θ with a, and b sin θ makes an angle 90° − θ with b. The solution is obtained by the intersections of the circles r=a cos θ + b sin θ, r=c.