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Projection Vector
Projection vector gives the projection of one vector over another vector. The vector projection is a scalar value. The vector projection of one vector over another is obtained by multiplying the given vector with the cosecant of the angle between the two vectors.
Vector Projection has numerous applications in physics and engineering, for representing a force vector with respect to another vector. Let us learn more about vector projection formula, derivation, and also check the examples.
What is Projection Vector?
The vector projection of one vector over another vector is the length of the shadow of the given vector over another vector. It is obtained by multiplying the magnitude of the given vectors with the cosecant of the angle between the two vectors. The resultant of a vector projection formula is a scalar value.
Projection of vector \(\vec a\) over vector \(\vec b \) is obtained bythe product of vector a with the Cosecant of the angle between the vectors a and b. This is further simplified to get the final value of the projection vector.The projection vector has a magnitude which is part of the magnitude of vector a, and has the direction same as the vector b.
Projection Vector Formula
The projection vector formula in vector algebra for the projection of vector a on vector b is equal to the dot product of vector a and vector b, divided by the magnitude of vector b. The resultant of the dot product is a scalar value, and the magnitude of vector b is also a scalar value. Hence the magnitude and argument of the projection vector answer is a scalar value, and is in the direction of vector b..
\(\text{Projection of Vector } {a} \ \text{on Vector } {b} = \dfrac{a. b}{ b}\)
Derivation of Projection Vector Formula
The following derivation helps in clearly understanding and deriving the projection vector formula for the projection of one vector over another vector. Let OA = \(\overrightarrow a\), OB = \(\overrightarrow b\), be the two vectors and θ be the angle between \(\overrightarrow a\) and \(\overrightarrow b\). It is the component of vector a across vector b. Draw AL perpendicular to OB.
From the right triangle OAL , cos θ = OL/OA
OL = OA cos θ
OL = \(\overrightarrow a\) cos θ
OL is the projection vector of vector a on vector b.
\(\overrightarrow a. \overrightarrow b\) = \(\overrightarrow a\overrightarrow b\) cos θ
\(\overrightarrow a. \overrightarrow b\)= \(\overrightarrow b(\overrightarrow a\) cos θ)
\(\overrightarrow a. \overrightarrow b\) = \(\overrightarrow b\) OL
OL = \(\dfrac{\overrightarrow a. \overrightarrow b}{\overrightarrow b}\)
Thus, projection vector formula of of vector \(\overrightarrow a\) on vector \(\overrightarrow b = \dfrac{\overrightarrow a. \overrightarrow b}{\overrightarrow b}\). Similarly, the projection of vector \(\overrightarrow b\) on vector \(\overrightarrow a = \dfrac{\overrightarrow a. \overrightarrow b}{\overrightarrow a}\).
Concepts Relating to Projection Vector
The following concepts below help in a better understanding of the projection vector. Let us check the details and the formula to find the angle between two vectors and the dot product of two vectors.
Angle Between Two Vectors
The angle between two vectors is calculated as the cosine of the angle between the two vectors. The cosine of the angle between two vectors is equal to the dot product of the individual constituents of the two vectors, divided by the product of the magnitude of the two vectors. The formula for the angle between the two vectors is as follows.
\(cos\theta = \dfrac{\overrightarrow a.\overrightarrow b}{a.b}\)
\(cos\theta = \dfrac{a_1.b_1 + a_2.b_2 +a_3.b_3}{\sqrt{a_1^2 + a_2^2 +a_3^3}.\sqrt{b_1^2 + b_2^2 + b_3^2}}\)
Dot Product of Two Vectors
The dot product between two vectors is also referred as scalar product. If the two vectors are expressed in terms of unit vectors, i, j, k, along the x, y, z axes, then the scalar product is obtained as follows:
If \(\overrightarrow a = a_1\hat i + a_2 \hat j + a_3 \hat k\) and \(\overrightarrow b = b_1 \hat i + b_2 \hat j + b_3\hat k\), then
\(\overrightarrow a. \overrightarrow b\) = \((a_1 \hat i + a_2 \hat j + a_3 \hat k)(b_1 \hat i + b_2 \hat j + b_3 \hat k)\)
\(\overrightarrow a. \overrightarrow b\) = \(a_1b_1\) + \(a_2b_2\)+ \(a_3b_3\)
Related Topics
The following topics help in a better understanding of the projection vector.
Examples on Projection Vector

Example 1: Find the projection of the vector \(4\hat i + 2\hat j + \hat k\) on the vector \(5\hat i 3\hat j + 3\hat k\), using the projection vector formula.
Solution:
Given \(\vec A = 4\hat i + 2\hat j + \hat k\) and \(\vec B = 5\hat i 3\hat j + 3\hat k\).
The projection vector formula of \(\vec A \) with respect to \(\vec B\). is as follows.
\(\begin{align}\text{Projection of Vector} \vec {A} \ \text{on Vector} \vec{B} = \dfrac{\vec{A}. \vec{B}}{ \vec{B}}\\&=\dfrac{(4.(5) + 2(3) + 1.(3))}{\sqrt{5^2 + (3)^2 + 3^2}}\\&=\dfrac{17}{\sqrt{43}}\end{align}\) 
Example 2: Find the projection of the vector \(5\hat i + 4\hat j + \hat k\) in the direction of the vector \(3\hat i + 5\hat j 2\hat k\), by using the projection vector formula.
Solution:
Given \(\vec A = 5\hat i + 4\hat j + \hat k\) and \(\vec B = 5\hat i 3\hat j + 3\hat k\).
\(\begin{align}\text{Projection of Vector} \vec {A} \ \text{on Vector} \vec{B} = \dfrac{\vec{A}. \vec{B}}{ \vec{B}}\\&=\dfrac{(5.(3) + 4(5) + 1.(2))}{\sqrt{3^2 + 5^2 + (2)^2}}\\&=\dfrac{33}{\sqrt{38}}\end{align}\)
FAQs on Projection Vector
What Is Projection Vector?
The projection vector is the shadow of one vector over another vector. The vector projection of one vector over another is obtained by multiplying the given vector with the cosecant of the angle between the two vectors.
What Is Projection Vector Formula?
The vector projection formula gives the projection of vector a on vector b. The projection vector formula is \(\text{Projection of Vector } \vec {a} \ \text{on Vector } \vec{b} = \dfrac{\vec{a}. \vec{b}}{ \vec{b}}\). The projection vector formula representing the projection of vector a on vector b is equal to the dot product of the two vectors, divided by the magnitude of the vector b.
How To Find Projection Vector?
Projection of vector \(\vec a\) over vector \(\vec b \) is obtained byproduct of vector a with the Cosecant of the angle between the vectors a and vector b. This can be further simplified to give the following formula of projection vector.
\(\text{Projection of Vector } \vec {a} \ \text{on Vector } \vec{b} = \dfrac{\vec{a}. \vec{b}}{ \vec{b}}\).
What Is Required To Find the Projection Vector?
To find the projection vector we need two vectors and the angle between the two vectors. Also, we need to know the vector for which the projection vector is to be calculated, with respect to the other vector.
How To Find the Angle Between the Two Vectors From the Projection Vector?
The angle between the two vectors can be computed by dividing the projection of a vector with the given vector. The two vectors are \(\vec a\) and \(\vec b \), and the projection of the vector \(\vec a\) is aCosθ. Thus the division of this projection with the magnitude of the vector gives the Cosecant of the angle between the two vectors.
How Do You Use Dot Product To Find the Projection Vector?
The for formul for the projection of \(\vec a\) over vector \(\vec b \) is \(\text{Projection of Vector } \vec {a} \ \text{on Vector } \vec{b} = \dfrac{\vec{a}. \vec{b}}{ \vec{b}}\). Here in the formula we observe the dot product of the \(\vec a\) and \(\vec b \) in the projection formula.
What Are the Applications of Projection Vector?
Vector Projection has numerous applications in physics and engineering, for representing a force vector with respect to another vector. The force applied at an angle has limited influence in the required direction. The component of this projection vector gives the exact applied force in the required direction.
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