Vector Projection Formula

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.

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