Direction cosines of a line:
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If a, b, g be the angles that a given directed line makes with the positive directions of co-ordinate axes, then cosa, cosb, cosg are called as direction cosines of given line and are denoted by l, m, n respectively.
Therefore, l = cosa, m = cosb and n = cosg
By definition it follows that direction cosine of the axis of x are respectively cos0°, cos90°, cos 90° i.e. (1, 0, 0).
Similarly direction cosines of axes of y and z are respectively (0, 1, 0) and (0, 0, 1).
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Relation between the Direction Cosines:
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Let OP be any line through origin O which has direction cosines l, m, n.
Let P ≡ (x, y, z) and OP = r
Then OP2 = x2 + y2 + z2 = r2 .... (1)
From P draw PA, PB, PC perpendicular on coordinate axes, so that
OA = x, OB = y, OC = z.
Also, ∠POA = a, ∠POB = b and ∠POC = g.
From triangle AOP, l = cosa = x/r => x = lr
Likewise y = mr and z = nr
Hence from (1)
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r2(l2 + m2 + n2) = x2 + y2 + z2 = r2 => l2 + m2 + n2 = 1
Note:
- If coordinates of any point P be (x, y, z) and l, m, n be direction cosines of the line OP, O being origin, then (lr, mr, nr) will give us co ordinates of a point on the line OP which is at the distance r from (0, 0, 0).
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