Dot Product

PROPERTIES OF SCALAR PRODUCT

a) The dot product between two vectors obeys commutative law i.e., \( \overrightarrow P .\overrightarrow Q = \overrightarrow Q .\overrightarrow P \)
b) The dot product obeys distributive law, i.e., \( \overrightarrow P .\left( {\overrightarrow Q + \overrightarrow R } \right) = \overrightarrow P .\overrightarrow Q + \overrightarrow P .\overrightarrow R \)
c) It is always a scalar which is positive if angle between the vectors is acute (i.e., <90⁰) and negative if angle between them is obtuse (i.e.,\( 90^0 < \theta \leqslant 180^0 \) )
d) Scalar product of two vectors will be maximum when , vectors are parallel.\( \Rightarrow \left( {\overrightarrow P .\overrightarrow Q } \right)_{\max } = PQ \)
e)    \( \overrightarrow P .\overrightarrow Q = P\left( {Q\cos \theta } \right) = Q\left( {P\cos \theta } \right) \) 
f) If the scalar product of two non zero vectors vanishes, then the vectors ar othogonal (perpendicular)
    i.e., \( \theta = 90^0 ,\overrightarrow P .\overrightarrow Q = PQ\cos 90^0 = 0 \)
g) The scalar product of a vector by itself is termed as self dot product and is given by \( \overrightarrow P .\overrightarrow P = PP\cos \theta = P^2 \left( {\theta = 0^0 } \right) \)
h) In case of unit vector \(\hat n\),
           \( \hat n .\hat n = 1 \times 1 \times \cos \theta = 1 \) 
             \( \hat n.\hat n=\hat i.\hat i=\hat j.\hat j=\hat k.\hat k=1 \)
i) In case of orthogonal unit vectors \(\hat i.\hat j\) and \(\hat k.\hat i.\hat j=\hat j.\hat k=\hat k.\hat i=0\)
j) If \( \overrightarrow P = P_X \hat i + P_Y \hat j + P_Z \hat k \) and \( \overrightarrow Q = Q_X \hat i + Q_Y \hat j + Q_Z \hat k \) 
             \( \therefore \overrightarrow P .\overrightarrow Q = \left( {P_X \hat i + P_Y \hat j + P_Z \hat k } \right).\left( {Q_X \hat i\ + Q_Y \hat j + Q_Z \hat k } \right) \) 
          \( = \left( {P_X Q_X + P_Y Q_Y + P_Z Q_Z } \right) \)
k) The product of vectors is possible between vectors of different kind.