# Scalar and Vector Projection Formula

Last Updated : 21 Dec, 2023

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

### ProjectionÂ

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

Â

Scalar projectionÂ  Â

Projection of

Similarly,Â

Projection of

Vector projectionÂ

Vector projection is defined as the product of scalar projection ofÂ Â onÂ Â and the unit vector along. Vector projection of

Similarly,Â

Vector projection of

### Sample Problems

Problem 1: IfÂ [Tex]Â Â Â Â Â  [/Tex]Â andÂ [Tex]Â Â Â Â Â  [/Tex]. then find the projection ofÂ a on b vector.

Solution : Â

Here,Â Â  andÂ Â  Â  Â  Â  Â  Â  Â Projection ofÂ Â =Â Â  Â  Â  Â  Â  Â  Â

Projection ofÂ Â =Â Â  Â  Â  Â  Â  Â Â

=Â

Problem 2: Find the projection of a vectorÂ a + b on c vector, here,Â Â  ,Â andÂ

Solution:Â

Here,Â Â  ,Â andÂ Â  Â  Â  Â  Â

Â Â  Â  Â  Â  Â  Â

Projection of vectorÂ Â onÂ Â  Â  Â  Â Â

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

Solution: Â

LetÂ Â andÂ Â  Â  Â  Â  Â  Â Â

Projection of Â Â =Â Â  Â  Â  Â  Â  Â Â

=Â Â = Â 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: Â

LetÂ Â andÂ Â  Â  Â  Â  Â  Â

Projection of Â Â =Â Â  Â  Â  Â  Â  Â Â

= 5/6

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

Solution :Â

Here, Â Projection ofÂ Â  Â  Â  Â  Â

Â and Â Â Â  Â  Â  Â  Â  Â

Projection ofÂ Â onÂ Â =Â Â  Â  Â  Â  Â  Â

4 Â =Â Â  Â  Â  Â  Â  Â

4 =Â Â  Â  Â  Â  Â  Â

28 = 2Î» + 18 Â  Â  Â  Â  Â Â

Î» = 5

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

Solution: Â

=Â Â andÂ Â  Â  Â  Â  Â  Â  Â  Â Â

Projection of the vectorÂ Â  Â  Â  Â  Â  Â  Â  Â

=Â

Problem 7: Find the vector projection of m on n vector, hereÂ Â andÂ

Solution: Â

Here,Â Â andÂ Â  Â  Â  Â  Â  Â Â

Vector projection=Â Â  Â  Â  Â  Â  Â Â

=Â Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â

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