Scalar and Vector Projection Formula

In vector algebra, projection means finding how much of one vector lies in the direction of another vector. It helps us understand the effect of one vector along another and is used in many problems of mathematics and physics. In vector algebra, projection means resolving one vector along the direction of another vector. Projection helps us understand how much of one vector lies in the direction of another.

There are two types of projections:

• Scalar Projection
• Vector Projection

1. Scaler Projection of Vector

Scalar projection of vector \overrightarrow{\rm a} on \overrightarrow{\rm b} is the length (magnitude) of the component of a in the direction of b. If θ is the angle between vector a and b, then Scalar projection of a on b =  |a| cosθ

\boxed {\text{Scaler Projection of } \overrightarrow{a} \text{ on } \overrightarrow{b}= \frac{\overrightarrow{a} \cdot \overrightarrow{b}}{|\overrightarrow{b}|}}

Similarly, 

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

2. Vector Projection of Vector

The vector projection of vector \overrightarrow{\rm a} onto vector \overrightarrow{\rm b} (written as \text{proj}_{\overrightarrow{\rm b}}\overrightarrow{\rm a}) is the orthogonal projection of \overrightarrow{\rm a} onto a straight line parallel to \overrightarrow{\rm b}.

\boxed{\text{Vector Projection} = \text{proj}_{\overrightarrow{\rm b}}\overrightarrow{\rm a}= \left(\frac{\overrightarrow{\rm a}\cdot \overrightarrow{\rm b}}{\|\overrightarrow{\rm b}\|^{2}}\right)\overrightarrow{\rm b}}

It represents the component of \overrightarrow{\rm a} that acts in the direction of \overrightarrow{\rm b}

Solved Examples

Problem 1: If \overrightarrow{a} = 7\hat{i} + \hat{j} -4\hat{k} and \overrightarrow{b} = 2\hat{i} + 6\hat{j} + 3 \hat{k}. 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 b + c on vector a , 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

Unsolved Questions

Problem 1: 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}  

Problem 2: 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}