Ballantine 1952
Ballantine 1952
Ballantine 1952
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CLASSROOM NOTES
EDITED BY G. B. THOMAS, MassachusettsInstituteofTechnology
All material
forthisdepartmentshouldbesenttoG. B. Thomas,Department
ofMathe-
matics,Massachusetts
Institute
of Technology,
Cambridge 39, Mass.
DISTANCE FROM A LINE, OR PLANE, TO A POINT*
J. P. BALLANTINE and A. R. JERBERT, University
ofWashington
""'8
IAI
IBI
To find the perpendicular distance from the line Ax+By+C=O, to the
point P(x1, yO),we begin by computingthe y-distance,
(1) MP= Yl-y2 = y - (-Axi-C)/B = (Ax, + Byl + C)/B,
where Y2 is the y-value obtained by substitutingx=xl in the equation of the
line. Since the y and x interceptsof the latter are in the ratio (-CB)(- C/A)
=A/B, the intercept triangle has sides proportionalto IA, |B A2+B2. |
MRP is evidentlya similar triangleso that,
RP/MP = B I //A2 + B2,
whence,
d =RP=- -MP
VA2 + B2
B (Ax, + By,+C) by equation (1),
BV/A2 + B2
+ + Cl
(2) Axi Byi (signum B =
B I /B).
(signum B)V-\A2+ B2
Any KOO, which is multipliedinto the coefficients A, B, C of the equation
of the line evidently "divides out" of the factorsin the rightmemberof equa-
tion (2). These factorsand theirproduct d are thereforeinvariant under such a
multiplication.The second factorMP, yields + and - values forpoints above
and below the line, respectively,and this remains true for d since the first
factor, I B V\A2+B2, is always positive.
* Thispaperis basedon Professor
J.P. Ballantine'streatmentofthedistanceformttlaon page
235 ofhisbookEssentialsofEngineeringMathematics, publishedby PrenticeHall, 1938.
242
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All use subject to JSTOR Terms and Conditions
1952] CLASSROOM NOTES 243
We now equate the rightsides of (2) and (4) canceling the common factor
x - y. In the resultingidentityit is still permissibleto make x = y. Hence,
(5) n(1 + X)8 1 = nCl + 2nC2x+ 3nC3x2+ * + n
nCnxn1.
On the other hand, because of (3), we also have
(6) n(1 + X)8 1 = n(1 + n-1C1X + n 1C2x2+ * + n-lcn-1 ).
On comparingthe rightsides of (5) and (6) we see that nCl = n. Then necessar-
ofx are equal we findnext nC2= n(n - 1)/2.
ily n-1C,= n-1. Since the coefficients
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