I am to prove that angle between two lines which intersect at origin is $\cos^{-1} (l_1 l_2 +m_1m_2 +n_1 n_2)$ . Here $(l_! , m_1 , n_1) , (l_2 , m_2 , n_2)$ are direction ratios of these two lines .
In the proof of it my book has supposed that projection of a line segment X on a line L is sum of projections of $A , B , C$ on the line L. Here $A, B , C$ are the projections of the line segment X on the Co-ordinate axes. I can not understand how it is happening .
Can anyone please tell me how to prove the bold part?
We have that projection of $\vec u$ onto $\vec v $ is given by
$$u_v=\vec u \cdot \frac{\vec v}{|\vec v|}$$
and by distributive property of dot product assuming $\vec u=A \hat i+B\hat j+C\hat k$
$$u_v=\vec u \cdot \frac{\vec v}{|\vec v|}=\left(A \hat i+B\hat j+C\hat k \right)\cdot \frac{\vec v}{|\vec v|}=A\, \hat i\cdot \frac{\vec v}{|\vec v|}+B\,\hat j\cdot \frac{\vec v}{|\vec v|}+C\,\hat k\cdot \frac{\vec v}{|\vec v|}$$