( vii ) ( a * b ) 2 ≤ | a | 2 2 | b | 2
( viii ) If a = a1i + a2j + a3k , then | a | 2 = a * a = a1 2 + a2 2 + a3 2 Or
| a | = √a1 2 + a2 2 + a3 2
( ix ) Angle between Two Vectors If θ is angle between two non-zero vectors , a , b , then we have
a * b = | a | | b | cos θ cos θ = a * b / | a | | b |
If a = a1i + a2j + a3k and b = b1i + b2j + b3k Then , the angle θ between a and b is given by cos θ = a * b / | a | | b | = a1b1 + a2b2 + a3b3 / √a1 2 + a2 2 + a3 2 √b1 2 + b2 2 + b3 2
( x ) Projection and Component of a Vector Projection of a on b = a *
b / | a | Projection of b on a = a * b / | a | Vector component of a vector a on b
Similarly , the vector component of b on a = (( a * b ) / | a 2 |) * a
( xi ) Work done by a Force