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Vectors and 3-D Geometry
   Introduction to Vectors and 3-D Geometry
   BASIC VECTOR OPERATIONS
   ADDITION OF VECTORS- TRIANGLE PARALLELOGRAM LAW
   SUBTRACTION OF VECTORS
   Example problems on ADDITION OF VECTORS
   MULTIPLICATION OF A VECTOR BY A SCALAR
   Example problems on MULTIPLICATION OF VECTOR
   linearly independent and dependent vectors
   RESOLUTION OF A VECTOR IN A GIVEN BASIS
   MAGNITUDE, DIRECTION COSINES AND DIRECTION RATIOS
   Example problems on vectors and COLLINEARITY
   Example problems on vectors and section formula
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MAGNITUDE, DIRECTION COSINES AND DIRECTION RATIOS

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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Posted By :-
 palinder18@gmail.com
Nov 15, 5:43:12 PM 0 4932

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