In vectors chapter, direction cosines are cosines of angles a line makes with axes, and direction ratios are proportional numbers like a,b,c. The sum of squares of direction cosines equals one because it comes from cos²α + cos²β + cos²γ = 1, derived from the fundamental identity. This feels clear on paper, but when I help my father calculate the exact proportions for his jadoh-style momo spice mix, the theory of perfect ratios meets the practical need to adjust for the day's humidity—so I trust the formula, but I also know some things need a little *sngewbha* beyond the textbook.
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