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Direction cosines

If α, β, γ are the angles which a vector 87088.png makes with the positive directions of the coordinates axes OX, OY, OZ respectively, then cos α, cos β, cos γ are known as direction cosines of 87094.png and are generally denoted by letter l, m, n respectively.
Thus, l = cos α, m = cos β, n = cos γ, where 0 ≤ α, β, γπ. Also l2 + m2 + n2 = 1.
Direction cosines of the x-axis, y-axis, and z-axis are (1, 0, 0), (0, 1, 0), and (0, 0, 1) respectively.
Any vector 87102.png and 87108.png a unit vector.

Direction ratios

Let l, m, n be the direction cosines of a vector 87115.png and a, b, c be three numbers such that a, b, c are proportional to l, m, n, i.e.,
87121.png or (l, m, n) = (ka, kb, kc)
(a, b, c) are direction ratios
There can be infinitely many direction ratios for a given vector, but the direction cosines are unique.


  • Direction ratios of the line joining two given points (x1y1z1) and (x2y2z2) is given by (x2 – x1y2 – y1z2 – z1).
  • The projection of segment joining the points P(x1y1z1) and Q(x2y2z2) on a line with direction cosines lmn is (x2 –x1)l + (y2 – y1)m + (z2 – z1)n.
  • If l1m1n1 and l2m2n2 are the direction cosines of two concurrent lines, then the direction cosines of the lines bisecting the angles between them are proportional tol1 ± l2m1 ± m2n1 ± n2.

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