II. EXTERNAL DIVISION OF A LINE
Suppose M- is the point which divides the line joining the points 
and 
where position vectors are 
respectively , externally is the ration 
=
……………(i)
……………(ii)
By using ratio theorem
i.e
BY CROSSING MULTIPLICATION
Therefore
The co-ordinate of M
External division of a line where 
QUESTIONS
1.Find the length of the line 
of 
2. Find the position vector which divides line 
having point 
into two equal points.
3. A and B are two points whose vectors are 3
+ 
and 
respectively. Find the position vector of the points dividing AB.
(a) Internally in the ratio 1:3
(b) Externally in the ratio 3:1
III. MULTIPLICATION OF A VECTOR
(A) SCALAR MULTIPLICATION OF A VECTOR
In this case a vector is multiplied by a certain constant called scalar
Let
THEREFORE
QUESTIONS
If 
(a) Find (i) 
(ii)
.
(b) Comment on results in 
above
DEFINITION OF DOT PRODUCT
For vectors 
Where
is the above between 
cos Q
cos QCHARACTERISTICS
1. 1. PARALLEL VECTOR
Two vector are said to be parallel of the angle between them is zero
Mathematically
From
= cos Q
But Q = 
= cos Q
This is one among the
2. 2. Orthogonal vectors
Two vectors are said to be orthogonal of the angle between them is 90
Mathematically
From
But Q = 90
(orthogonal or perpendicular vector)
= cos 90
This is conditional for the orthogonal vector