Vector Projection Formula

An Overview of Vector Projection

A vector is an object that has both magnitudes as well as direction. Two examples of vectors are both forces as well as velocity. The magnitude of the force indicates the strength, while if we talk about the magnitude of velocity it indicates the speed. Along with this, it gives you direction also.


The vector projection is the vector that is produced when one vector is just divided into two vectors. In vectors that are divided, one vector is parallel to the other vector and another vector is perpendicular to the given vector. The parallel vector is the vector projection. Any work that will be done across this direction will be easily done as compared to the work done in any other direction. In this process, you are actually producing an actual vector instead of any scalar quantity. There is a fact that if you apply force in the direction in which the vector is divided then the force you will need to apply will be less as compared to the actual force that will be applied in any other direction.


If we talk about the definition of its scalar projection, then instead of direction only magnitude will be there. Remember when the scalar projection is positive, the angle is going to be less than 90° and when the scalar quantity will be negative, it is going to be more than 90°. This means that both the vectors are going in opposite directions.


There is also a difference in the font that these vectors use. The font will be bold for vector value and normal for scalar value but when it comes to the handwritten form, then there is a presentation of an arrow on the quantity that denotes vector and for the one that denotes scalar quantity, it is going to be simple without any arrows. Now let's have a brief discussion about this vector projection formula, properties of vector projection and finally, what we conclude from it.


What is the Formula for Vector Projection?

A quantity with magnitude and direction represented with an arrow over its symbol is a vector. When we displace a body parallel to itself, the vector does not change. Though it has magnitude and direction, it does not have a specific location. Projection of a vector a on another non-zero b vector is the orthogonal projection of the first vector on a straight line parallel to the second vector. Projection of vector a on b formula can be denoted by projba.


Formula for Vector Projection

Vector projection is defined for a vector when resolved into its two components of which one is parallel to the second vector and one which is perpendicular to the second one. The force that is applied across this vector will be less and the work can be easily done if we do it in this direction. We will define the vector projection formula with the help of two vectors, say a and b. Consider the below diagram:

In the above diagram, there are two vectors, a and b, and θ is the angle between them. Then vector projection is given by:

\[proj_{b}a=\frac{\vec{a}\cdot \vec{b}}{b^{2}}\vec{b}\]

In the above diagram ‘.’ operation defines a dot product between vectors a and b.

The scalar projection of a vector a on b is given by: 

\[a_{1}\left \| a \right \|cos\Theta\]

Here θ is the angle that a vector a makes with another vector b. a1 is the scalar factor.

Also, vector projection is given by

\[a_{1}=a_{1}b = (\left \| a \right \|Cos\Theta)b\]

The projection of a vector a on b formula gives a vector having the direction of vector b.


Vector Projection Equation in Terms of a and b

If we are not defined with a value of θ, we can write a vector projection equation in terms of a and b. A dot product formula is used in such cases given by:

\[\frac{a.b}{\left \| a \right \|\left \| b \right \|}=cos\theta \]

Hence substituting the value of cosθ in the scalar projection, we get:

\[a_{1}=\left \| a \right \|cos\theta =\left \| a \right \|\frac{a.b}{\left \| a \right \|\left \| b \right \|}=\frac{a.b}{\left \| b \right \|}\]

Similarly substituting in vector projection, we get:

\[a_{1}=a_{1}\hat{b}=\frac{a.b}{\left \| b \right \|}\frac{b}{\left \| b \right \|}\]


Properties of Vector Projection Formula

According to the above vector projection equation, there are certain defined properties on it. Considering θ as the angle between two vectors, the projection properties are given below:

When θ is 90° a1 will be 0.  In this case, your vector is going to have a positive value.

If 90° < θ ≤ 180° b and a1 have opposite directions. In this case, the vector is going to have a negative value.

If 0 ≤ θ < 90° a1 and vector b have the same direction. In this case, you are going to get a parallel vector.

The concept of vector projection has high utilisation in Gram–Schmidt orthonormalization. Also, the concept is helpful to detect if two convex shapes intersect each other or not.


Conclusion

While solving different problems of vector projection, make sure you are well aware of two different formulas stated above. The formula for vector projection is situational and dependent upon the angle between two different vectors.


For brief information on the vector topic, you can visit Vedantu now. They have provided you with detailed information on the topics that will help you to get good knowledge and it is going to clear your basic concepts also so that it will be easy for you to score well in your examination. At Vedantu, our main motive is to increase the capability of the students so here we are providing you with the detailed information of all the topics so that your concept is cleared and along with this, you can score well and get a good grip on your topics. Some of the topics are the basic ones that are going to be the foundation of Physics as well as Maths. Vector is one such topic in which you need to get a good idea for examination point of view and so for that, you can visit the Vedantu app. Here, you will get the desired notes specially prepared by the experts.

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FAQs (Frequently Asked Questions)

1. How will you Define the Projection of a Vector?

One can define a vector as any quantity that has both magnitude and direction. When you divide a vector into two, the parallel vector is going to be the vector projection which requires minimal force to work easily. For a vector projection, if one vector is projected in the direction of another vector, we define it as an orthogonal projection of the vector component of the first vector in the direction of the second vector. It’s denoted by projba where a is the first vector projected over second vector b.

2. What is the Use of Concept Vector Projection?

The concept of vector projection is very basic. It is basically to make you understand and define components of a vector in the direction of another vector. In practical life, if you consider this topic of vector projection, then it can be helpful in detecting whether two convex shapes are intersecting or not. These concepts such as that of vector and its projection have uses in theoretical as well as practical life as we have considered in the above example. Similarly, you are going to get such examples on a daily basis every day.

3. How is Scalar Projection Different From Vector Projection?

If you talk about scalar projection, then it is mainly considered with the magnitude and there is no use of direction in it but in the same way, if we talk about vectors then besides considering the magnitude for vectors, you need to consider the direction. Instead, you can say that scalar projection is the magnitude of the resultant projection, which also defines the length of vector projection. Vector projection has the magnitude and direction of a vector. Vector projection gives results in the form of a vector. The magnitude of vector projection is the scalar projection of a vector.

4. What is required to find the projection of a vector?

To find the projection of a vector, we are required to have two vectors whose projection we need to find and we need the angle between the two vectors as well. Say if the vectors are having an angle less than 90 degrees, it is going to be a positive vector and if it is going to be more than 90 degrees then it will be a negative vector. Besides this, we need to know the vector whose projection vector we need to find with respect to the other vector.

5. What is the application of the projection of vectors?

In the field of Physics as well as engineering, the projection of the vector has multiple uses; like if you need to represent the force vector with respect to any other vector then the vector projection will be used. The component of the vector projection is going to give you the exact value of the applied force in the given direction. There are other uses of vector projection in the engineering and Physics fields also but the main use has been explained to you. It is the basic topic that is going to help you in future classes so make sure you clear your concept and for that, you can use the Vedantu app.

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