1. CE Statics
Lecture 6

2. Contents
Dot Product

3. DOT PRODUCT
Dot product is the method of multiplying two vectors and is used to solve three-dimensional problems
If A and B are two vectors
then,
A . B = AB cos ()
where 0   180  A B

4. Laws of Operation (1) Commutative Law A . B = B . A
(2) Multiplication by a scalar
a  ( A . B ) = ( aA ) . B = A . ( aB )
= ( A . B )  a
(3) Distribution Law
A . ( B + D ) = ( A . B ) + ( A . D )

5. Cartesian Vector Formulation
i . i = 1 j . j = 1 k . k = 1
i . j = 0 i . k = 0 k . j = 0
i . i = (1)  (1) cos 0 = 1
i . k = (1)  (1) cos 90 = 0
If A and B are Cartesian vectors, then
A . B = (Ax i + Ay j + Az k) . (Bx i + By j + Bz k
= Ax Bx (i.i) + Ax By (i.j) + Ax Bz (i.k)
Ay Bx (j.i) + Ay By (j.j) + Ay Bz (j.k)
Az Bx (k.i) + Az By (k.j) + Az Bz (k.k)
Doing Dot Product, we have
A . B = Ax Bx + Ay By + Az Bz [scalar ( so there is no i, j, k)]

6. Applications
(1) to find the angle  formed between two vectors or two intersecting lines.
If A and B are vectors, then  between their tails will be:
 = cos-1 ( A . B ) / (AB)    180
If A . B =  = cos-1 (0) = 90 (A is perpendicular to B)

7. to find the rectangular components of a vector. AII = A cos ()
AII = A cos  = A . u
AII = A cos () u = (A . u) u
AI = A – AII (since A = AII + AI)
To find the magnitude of AI
If cos () = A . U /A
 = cos-1 ( A . u ) / A (0   180)
AI = A sin ()
or AI = A2 – (AII)2 (Pythagorean) a a
AII = A cos () u AI A u 

8. Examples
Examples 2.16 – 2.17
Problem 2.113
Problem 2.129

15. Assignment No. 1 (Chapter 2)
1, 9, 12, 19, 23, 28, 29, 33, 36, 40, 47, 54, 58, 60, 65, 69, 73, 76, 78, 80