Consider a vector
 = xi + yj + zk
as shown in the figure below:
The magnitude or is simply the length of the diagonal of the cuboid whose sides are x, y and z. Thus
||= Ö(x2 + y2 + z2) ... (1)
Suppose makes angles a ,b andl with the X, Y and Z axis, as shown:
Then the quantities
l = cos a
m = cos b
n = cos g
are called the direction cosines of (abbreviated as DCs. The DCs uniquely determine the direction of the
vector. Note that since
= xi + yj + zk
we have
From (1), this gives
L2 + m2 + n2 =1
We can also infer from this discussion that the unit vector r along can be written as
Direction ratios (DRs) of a vector are simply three numbers, say a, b and c, which are proportional to the DCs, i.e
(1/a) + (m/b) =(n/c)
It follows that DRs are not uniqe (DCs obviously are)
From a set of DRs {a, b, c}, the DCs can easily be deduced:
Before we go on to solving examples involving the concepts we.ve seen till now, you are urged to once again go over the entire earlier discussion we.ve had, so that the .big picture. is clear in your mind.
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