defination of the measurement and vectors write it clearly

Scalar or dot product of two vectors a and b is defined as
a . b = | a | | b | cos θ where θ = ( a . b ) and 0 ≤ θ 180 o .
2. a . b = 0 ⇒ a = 0 or b = 0 or ( a . b ) = 90 o .
3. a . b = | a | | b | if ( a . b ) = 0.
4. a . b = b . a ; a . ( b + c ) = a . b + a . c .
5. If a . b = ± | a | | b | where a , b are two non-zero vectors, then they are parallel.
6. | a . m b | = lm( a . b ).
7. a . a = | a | 2 = a 2 .
8. If a . b > 0 then ( a . b ) 90 o .
9. Component of a on b (or) Orthogonal projection of a on b (scalar) =
( a . b )
.
| b |
10. Component of a on b (length) (or) Length of orthogonal projection of a on b =
11. Component vector of a on b (or) orthogonal projection of a on b (vector) =
| a . b |
.
| b |
( a . b ) b
.
| b | 2
12. Component vector of a perpendicular to b (or) orthogonal projection of a perpendicular to b
= a –
( a . b ) b
| b | 2
13. ( a + b ) . ( a – b ) = a 2 – b 2 .
14. ( a + b ) 2 = a 2 + 2 a . b + b 2 .