DIRECTION RATIO AND DIRECTION COSINES
I. DIRECTION RATIO
Suppose the vector
The direction ratio is given by
II. DIRECTION COSINE
Consider the vector
From three dimension plane.
Makes angles 
with 
direction respectively
Hence
Therefore the direction cosines are
FACT IN DIRECTION COSINES
– Suppose the vector
Also the direction cosines are
Hence
+
+
= 1
– The sum of square of the direction cosines is one.
Proof
i.e 
= x
+y
+z
=
Also
—————-(i)
———————-(ii)
—————————-(iii)
Adding the equation (i) , (ii) and (iii)
+
= 
+ 
= 
But
+
=
UNIT VECTOR
-Is the vector whose magnitude (modules) is one line a unit
-The unit vector in the direction of vector a is donated by read as ” a cap” thus
NOTE:
Any vector can be compressed as the product of it’s magnitude and it’s unit vector
i.e
QUESTIONS
1. Find a vector in the direction of vector 
which has a magnitude of 8 units
2. Find the direction ratio and direction cosines of the vector 
where p is the point (2, 3, -6)
THE FORMULA OF DISTANCE BETWEEN TWO POINTS
Suppose the line joining the points 
and 
whose position vectors are a and b respectively
HENCE
=
=
Hence
-Formula distance between two point
MID POINT OF A LINE
Suppose M is the point which divide the line joining the points 
and 
whose position vectors are respectively a and b into two equal parts
i.e
a = 
b = 
Hence
Therefore
The co-ordinate of M is
INTERNAL AND EXTERNAL DIVISION OF A LINE (RATIO THEOREM)
I. INTERNAL DIVISION OF A LINE
– Suppose M- is the point which divides the line joining the points 
and 
whose point vectors are a and b respectively internally in the ratio X:ee
a = 
=
b = 
=
……………(i)
……………(ii)
By using ratio theorem
By using multiplication
The ordinate form of M is
