In simple words, the cosines of the angles made by a directed line segment with the coordinate axes are called as the direction cosines of that line.
As illustrated in the figure above, if α, β, and γ are the angles made by the line segment with the coordinate axis then these angles are termed to be the direction angles and the cosines of these angles are the direction cosines of the line. Hence, cos α, cos β and cos γ are called as the direction cosines and are usually denoted by l, m and n.
l = cos α, m = cos β and n = cos γ
Another concept related to direction cosines is that of direction ratios. Three numbers which are proportional to the direction cosines of a line are called as the direction ratios. Hence, if ‘a’, ‘b’ and ‘c’ are the dr’s and l, m, n are the dc’s then, we must have
a/l = b/m = c/n.
This concept has been discussed in detail in the coming sections.
Let OP be any line through the origin O which has direction cosines l, m, n.
Let P be the point having coordinates (x, y, z) and OP = r
Then OP^{2} = x^{2} + y^{2} + z^{2} = r^{2} …. (1)
From P draw PA, PB, PC perpendicular on the coordinate axes, so that OA = x, OB = y, OC = z.
Also, ∠POA = α, ∠POB = β and ∠POC = γ.
From triangle AOP, l = cos α = x/r ⇒ x = lr
Similarly y = mr and z = nr
Hence from (1) r^{2 }(l^{2} + m^{2} + n^{2}) = x^{2} + y^{2} + z^{2} = r^{2}
⇒ l^{2} + m^{2} + n^{2} = 1
If we have two points P(x_{1}, y_{1}, z_{1}) and Q(x_{2}, y_{2}, z_{2}), then the dc’s of the line segment joining these two points are
(x_{2}x_{1})/PQ, (y_{2}y_{1})/PQ , (z_{2}z_{1})/PQ
i.e. (x_{2}x_{1})/√Σ(x_{2}x_{1})^{2}, (y_{2}y_{1})/√Σ(x_{2}x_{1})^{2}, (z_{2}z_{1})/√Σ(x_{2}x_{1})^{2 }
Example: A line with positive direction cosines passes through the point P(2, 1, 2) and makes equal angles with the coordinate axis. The line meets the plane 2x + y + z = 9 at point Q. The length of the line segment PQ equals
The direction cosines are l = m = n = 1/√3.
Hence, the equation of the required line is
(x2)/(1/√3) = (y+1)/(1/√3) = (z2)/(1/√3)
Hence, this gives x2 = y+1 = z2 = r
Hence, any point on the line is Q = (r+2, r1, r+2).
Since Q lies on the plane 2x + y + z = 9
Therefore 2(r+2) + (r1) + (r+2) = 9
This yields 4r + 5 = 9 or r = 1.
Hence teh coordinates of Q are (3,0,3).
Hence, PQ = √[(32)^{2} + (0+1)^{2} + (32)^{2}]
= √3
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