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Direction Ratios:

If a, b, c are three numbers proportional to the direction cosine l, m, n of a straight line, then a, b, c are called its direction ratios. They are also called direction numbers or direction components.

Hence by definition, we have

1/a = m/b = n/c = k (say)

=> l = ak, m = bk, n = ck => k2(a2 + b2 + c2) = l2 + m2 + n2 = 1

=> k = ± 1 / √a2 + b2 + c2 = ± 1/√Σa2

 l = ± a/√Σa2.   Similarly m = ± b/√Σa2 and n = ± n/√Σa2

where the same sign either positive or negative is to be chosen throughout.

Example: If 2, – 3, 6 be the direction ratios, then the actual direction cosines are 2/7, –3/7, 6/7.

Note:

Direction cosines of a line are unique but direction ratios of a line in no way unique but can be infinite.

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