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Scalar and Vector Projection Formula

Last Updated : 21 Dec, 2023
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Before Vector projection we have to look at scalar projection or generally we says projection of \overrightarrow{b} on \overrightarrow{a}      , means vector \overrightarrow{b}      produces projection on vector \overrightarrow{a}      . 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 \overrightarrow{a}      and these two vectors are close to each other from one side and make an angle θ in between them. Vector \overrightarrow{b}      makes projection on vector \overrightarrow{a}      . For better clarification, you can assume there are two sticks as like vector position. we put a torch in on condition over vector \overrightarrow{b}     . Then you see an shadow on first stick vector \overrightarrow{a}       that shadow is projection made by second stick (vector \overrightarrow{b}     ) on first stick (\overrightarrow{a}     ). 

 

Scalar projection   

Projection of \overrightarrow{a} on \overrightarrow{b} = \frac{ \overrightarrow{a}.\overrightarrow{b} } { |\overrightarrow{b}| }

Similarly, 

Projection of \overrightarrow{b} on \overrightarrow{a} = \frac{ \overrightarrow{a}.\overrightarrow{b} } { |\overrightarrow{b}| }

Vector projection 

Vector projection is defined as the product of scalar projection of \overrightarrow{a}      on  \overrightarrow{b}      and the unit vector along \overrightarrow{b}     . Vector projection of \overrightarrow{a} on \overrightarrow{b} = (\frac{\overrightarrow{a}.\overrightarrow{b}}{|\overrightarrow{b}|})\frac{\overrightarrow{b}}{|\overrightarrow{b}|} = \frac{(\overrightarrow{a}.\overrightarrow{b})\overrightarrow{b}}{|\overrightarrow{b}|^2}

Similarly, 

Vector projection of \overrightarrow{b} on \overrightarrow{a} = (\frac{\overrightarrow{a}.\overrightarrow{b}}{|\overrightarrow{b}|})\frac{\overrightarrow{a}}{|\overrightarrow{a}|} = \frac{(\overrightarrow{a}.\overrightarrow{b})\overrightarrow{a}}{|\overrightarrow{a}|^2}

Sample Problems

Problem 1: If \overrightarrow{a} = 7\hat{i} + \hat{j} -4\hat{k}     [Tex]      [/Tex] and \overrightarrow{b} = 2\hat{i} + 6\hat{j} + 3 \hat{k}     [Tex]      [/Tex]. then find the projection of a on b vector.

Solution :  

Here, \overrightarrow{a} = 7\hat{i} + \hat{j} -4\hat{k}       and \overrightarrow{b} = 2\hat{i} + 6\hat{j} + 3 \hat{k}                  Projection of \overrightarrow{a} on \overrightarrow{b}      = \frac{ \overrightarrow{a}.\overrightarrow{b} } { |\overrightarrow{b}| }                  

Projection of \overrightarrow{a} on \overrightarrow{b}      = \frac{  (7\hat{i} + \hat{j} - 4\hat{k}). ( 2\hat{i} + 6 \hat{j} + 3\hat{k} ) } { | 2\hat{i} + 6\hat{j} + 3\hat{k}|}                 

\frac{ 14+ 6 -12 }{√(4 + 36 + 9)} = \frac{8}{7}.

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

Solution: 

Here, \overrightarrow{a} = 2\hat{i} - 2\hat{j} + \hat{k}       ,\overrightarrow{b} =  \hat{i} + 2\hat{j} -2 \hat{k}      and \overrightarrow{c} = 2\hat{i} - \hat{j} + 4\hat{k}              

 \overrightarrow{b}  + \overrightarrow{c } = 3\hat{i} + \hat{j} + 2\hat{k}                

Projection of vector \overrightarrow{b}  + \overrightarrow{c }      on \overrightarrow{a} = \frac{ (3\hat{i} + \hat{j} + 2\hat{k} ).( 2\hat{i} - 2\hat{j} + \hat{k})}{|2\hat{i} - 2\hat{j} + \hat{k}|}             

= \frac{ 6 - 2 + 2 }{√(4+ 4+ 1)} = 6/3 = 2

Problem 3: Find the projection of the a on b vector, here, \overrightarrow{a} = \hat{i} -\hat{j}      and \overrightarrow{b}=\hat{i}+\hat{j}        

Solution:  

Let \overrightarrow{a} = \hat{i} -\hat{j}      and \overrightarrow{b}=\hat{i}+\hat{j}                 

Projection of  \overrightarrow{a} on \overrightarrow{b}      = \frac{(\hat{i} - \hat{j}).(\hat{i} +\hat{ j}) }{√2}                 

\frac{1-1}{√2}      =  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 \overrightarrow{a} = 2\hat{i} - \hat{j} + \hat{k}      and \overrightarrow{b} = \hat{i} -2\hat{j} +\hat{k}                

Projection of  \overrightarrow{a} on \overrightarrow{b}      = \frac{2 + 2 + 1 }{√6}                 

= 5/6

Problem 5: Find the value of λ when the scalar projection of a on b is 4, here, \overrightarrow{a} = λ\hat{i} + \hat{j} +4\hat{k}     \overrightarrow{b} = 2\hat{i} + 6\hat{j} + 3\hat{k }       

Solution : 

Here,  Projection of \overrightarrow{a} on \overrightarrow{b} = 4              

\overrightarrow{a} = λ\hat{i} + \hat{j} +4\hat{k}      and   \overrightarrow{b} = 2\hat{i} + 6\hat{j} + 3\hat{k }                

Projection of \overrightarrow{a}      on \overrightarrow{b}      = \frac{ \overrightarrow{a}.\overrightarrow{b} } { |\overrightarrow{b}| }                

4  = \frac{ 2λ + 6 + 12 } {√(4+ 36  + 3)}                

4 = \frac{2λ + 18 }{7}                

28 = 2λ + 18           

λ = 5

Problem 6: The projection of the vector a on b, here, \overrightarrow{a}     \hat{i}-2\hat{j}+2\hat{k}      and \overrightarrow{b}= 4\hat{i}-4\hat{j}+\hat{k}       

Solution:  

\overrightarrow{a}     \hat{i}-2\hat{j}+2\hat{k}      and \overrightarrow{b}= 4\hat{i}-4\hat{j}+\hat{k}                     

Projection of the vector \overrightarrow{a} on \overrightarrow{b} = \frac{4 + 8 + 2}{√33}                    

\frac{14}{√33}

Problem 7: Find the vector projection of m on n vector, here \overrightarrow{m} = \hat{i}-3\hat{j}+5\hat{k}      and \overrightarrow{n}= 4\hat{j}+3\hat{k}

Solution:  

Here, \overrightarrow{m} = \hat{i}-3\hat{j}+5\hat{k}      and \overrightarrow{n}= 4\hat{j}+3\hat{k}                 

Vector projection= \frac{(0 - 12 + 15)(4\hat{j} + 3\hat{K})} {√25}                 

\frac{12}{25}\hat{j}+\frac{9}{25}\hat{k}                              



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