Mathematical Methods for Physics and Engineering - Matematica.NET

7.7 EQUATIONS OF LINES, PLANES AND SPHERESRAbraOFigure 7.12 The equation of a line. The vector b is in the direction AR andλb is the vector from A to R.since R can be reached by starting from O, going along the translation vectora to the point A on the line and then adding some multiple λb of the vector b.Different values of λ give different points R on the line.Taking the components of (7.39), we see that the equation of the line can alsobewrittenintheformx − a xb x= y − a yb y= z − a zb z= constant. (7.40)Taking the vector product of (7.39) with b and remembering that b × b = 0 givesan alternative equation for the line(r − a) × b = 0.We may also find the equation of the line that passes through two fixed pointsA and C with position vectors a and c. SinceAC is given by c − a, the positionvector of a general point on the line isr = a + λ(c − a).7.7.2 Equation of a planeThe equation of a plane through a point A with position vector a and perpendicularto a unit position vector ˆn (see figure 7.13) is(r − a) · ˆn =0. (7.41)This follows since the vector joining A to a general point R with position vectorr is r − a; r will lie in the plane if this vector is perpendicular to the normal tothe plane. Rewriting (7.41) as r · ˆn = a · ˆn, we see that the equation of the planemay also be expressed in the form r · ˆn = d, or in component form aslx + my + nz = d, (7.42)227