Question

The projection of a vector on another vector is___________.

Answer 1

If a and b are two vectors separated by an angle φ , then the projection of the vector a on vector b which would be the same as the projection of vector b on vector a composes of a scalar component and a vector component

The scalar component is nothing but the length of the projection of a in the direction of b is the. In the below a_b denotes the scalar component of the projection of vector a on vector b.

As we know  

$cos\ cos\varphi = \frac{a_b}{\vec{|a|}}$  

$a_b=\vec{|a|} cos \varphi$  

Using the dot product formula $a_b=\vec{|a|}.\frac{\vec{a}.\vec{b}}{\vec{|a|}\vecc{|b|}}=\frac{\vec{a}.\vec{b}}{|b|}$, we get

 

The vector component of the projection of the vector a on vector b is defined as the product of the scalar component ab and the unit vector of the vector b $(\frac{\vec{b}}{|b|})$

Hence  

$\vec{a}_b=\frac{\vec{a}.\vec{b}} {\vec{|b|}}*\frac{\vec{b}} {\vec{|b|}} =\frac{\vec{a}.\vec{b}} {\vec{|b|}^2} * \vec{|b|}$