Scalar and Vector Projection Formula

Before Vector projection we have to look at scalar projection or generally we says projection of  , means vector  produces projection on vector . Projections are basically of two types: Scalar projections and vector projections.  Scalar projection tells us about the magnitude of the projection or vector projection tells us about itself and the unit vector of the projection.

Projection 

Let’s considered two vectors  and these two vectors are close to each other from one side and make an angle θ in between them. Vector  makes projection on vector  . For better clarification, you can assume there are two sticks as like vector position. we put a torch in on condition over vector . Then you see an shadow on first stick vector   that shadow is projection made by second stick (vector ) on first stick (). 

Scalar projection   

Projection of

Similarly, 

Projection of

Vector projection 

Vector projection is defined as the product of scalar projection of  on  and the unit vector along. Vector projection of

Similarly, 

Vector projection of

Sample Problems

Problem 1: If [Tex]      [/Tex] and [Tex]      [/Tex]. then find the projection of a on b vector.

Solution :  

Here,   and              Projection of  =              

Projection of  =             

= 

Problem 2: Find the projection of a vector a + b on c vector, here,   , and 

Solution: 

Here,   , and          

            

Projection of vector  on         

Problem 3: Find the projection of the a on b vector, here,  and    

Solution:  

Let  and             

Projection of   =             

=  =  0

Problem 4: Find the scalar projection of a on b, here, \overrightarrow{a} = 2\hat{i} – \hat{j} + \hat{k}, \overrightarrow{b} = \hat{i} -2\hat{j} +\hat{k}

Solution:  

Let  and            

Projection of   =             

= 5/6

Problem 5: Find the value of λ when the scalar projection of a on b is 4, here, ,   

Solution : 

Here,  Projection of          

 and              

Projection of  on  =            

4  =            

4 =            

28 = 2λ + 18           

λ = 5

Problem 6: The projection of the vector a on b, here, =  and   

Solution:  

=  and                 

Projection of the vector                

= 

Problem 7: Find the vector projection of m on n vector, here  and 

Solution:  

Here,  and             

Vector projection=             

=