Vector Projection Formula

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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. 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 dot product between vectors a and b.

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

\[a_{1} = ||a|| cos \theta\]

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

\[a_{1} = a_{1} \widehat{b} = (||a|| cos \theta )\widehat{b}\]

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 right vector projection equation in terms of a and b. A dot product formula is used in such cases given by:

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

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

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

Similarly substituting in vector projection, we get:

\[a_{1} = a_{1} \widehat{b} = \frac{a.b}{||b||} \frac{b}{||b||}\]


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.

  • If 90° < θ ≤ 180° b and a1 have opposite direction.

  • If 0 ≤ θ < 90° a1 and vector b have the same direction.

The concept of vector projection has high utilization 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.

FAQ (Frequently Asked Questions)

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

Ans: 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?

Ans: Vector projection equations are mainly used to define the component of one vector in another vector’s direction. In practical life, the concept is helpful to detect if two convex shapes are intersecting or not.

3. How is Scalar Projection Different From Vector Projection?

Ans: 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.