vector

Vectors 373
Example 1. If a and b be two unit vectors and be the angle between them, then find
the value of such that a + b is a unit vector. (Nagpur, University, Winter 2001)
Solution.Let
OA = a be a unit vector and
be the angle between a and b .
If
OB = c = a + b is also a unit vector then, we have
| OA | = 1
| OB | = 1
| OB | = 1
OAB is an equilateral triangle.
Hence each angle of OAB is 3
5.5 POSITION VECTOR OF A POINT
AB = b is another unit vector and
Ans.
The position vector of a point A with respect to origin O is the vector OA which is
used to specify the position of A w.r.t. O.
—
To find AB if the position vectors of the point A and point B are given.
If the position vectors of A and B are a and b . Let the origin be O.
Then
=
OA
a,
OB b
OA AB = OB
AB
= OB OA
AB
= b
O
a
AB
= Position vector of B – Position vector of A
Example 2. If A and B are (3, 4, 5) and (6, 8, 9), find AB
Solution.
AB
.
= Position vector of B – Position vector of A
= (6 iˆ 8 ˆj 9 kˆ) (3ˆi 4ˆj 5 kˆ)
= 3iˆ
4 ˆj 4 kˆ
Ans.
5.6 RATIO FORMULA
To find the position vector of the point which divides the line joining two given
points.
Let A and B be two points and a point C divides AB in the ratio of m : n.
Let O be the origin, then
OA
=
A
m C n B
a , and OB b, OC ? (a)
(b)
OC
= OA AC
m
m
= OA AB AC
AB
m n m n
m
= a .( b a ) ( AB b a)
m n
O
a
A
c
a + b = c
B (b)
O
b
B
A (a)