Which statement best distinguishes direction ratios from direction cosines for a line in three dimensions?
ADirection ratios are always unique, but direction cosines are not
BDirection cosines can be any three real numbers, but direction ratios must satisfy a unit-sum condition
CDirection ratios are any numbers proportional to a direction vector, while direction cosines are the normalized components satisfying $l^2+m^2+n^2=1$
DThere is no difference; the two terms always mean exactly the same ordered triple
Answer & Solution
Correct answer: C. Direction ratios are any numbers proportional to a direction vector, while direction cosines are the normalized components satisfying $l^2+m^2+n^2=1$
Direction ratios are any nonzero proportional components of a vector parallel to the line, so many triples can represent the same direction. Direction cosines are the cosines of the angles with the coordinate axes, so they are the normalized components of a unit direction vector and satisfy $l^2+m^2+n^2=1$. That is why option C is correct, while A and D ignore non-uniqueness and normalization.
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