Vector Projection Formula Derivation
The vector projection formula derivation is discussed below:
Let us assume, OP = [Tex]\vec A[/Tex] and OQ = [Tex]\vec B[/Tex] and the angle between OP and OQ is θ. Drawn PN perpendicular to OQ.
In the right triangle OPN, Cos θ = ON/OP
⇒ ON = OP Cos θ
⇒ ON = |[Tex]\vec A[/Tex]| Cos θ
ON is the projection vector of [Tex]\vec A[/Tex] on [Tex]\vec B [/Tex]
[Tex]\vec A.\vec B = |\vec A||\vec B|cos \theta [/Tex]
⇒ [Tex]\vec A.\vec B = |\vec B(|\vec A||cos \theta) [/Tex]
⇒ [Tex]\vec A.\vec B = |\vec B|ON [/Tex]
⇒ ON = [Tex]\frac{\vec A.\vec B}{|\vec B|} [/Tex]
Hence, the ON = [Tex]|\vec A|.\hat B [/Tex]
Thus the Vector Projection of [Tex]\vec A[/Tex] on [Tex]\vec B[/Tex] is given as [Tex]\frac{\vec A.\vec B}{|\vec B|} [/Tex]
the Vector Projection of [Tex]\vec B[/Tex] on [Tex]\vec A[/Tex] is given as [Tex]\frac{\vec A.\vec B}{|\vec A|} [/Tex]
Also Check: Types of Vectors
Vector Projection – Formula, Derivation & Examples
Vector Projection is the shadow of a vector over another vector. The projection vector is obtained by multiplying the vector with the Cos of the angle between the two vectors. A vector has both magnitude and direction. Two vectors are said to be equal if they have the same magnitude as well as the direction. Vector Projection is essential in solving numerical in physics and mathematics.
In this article, we will learn about what is vector projection, the vector projection formula example, the vector projection formula, vector projection formula derivation, vector projection formula linear algebra, vector projection formula 3d, and some other related concepts in detail.
Table of Content
- What is Vector Projection?
- Vector Projection Formula
- Vector Projection Formula Derivation
- Vector Projection Formula Examples
- Practical Applications and Significance of Vector Projection
- Real-World Problem-Solving Examples of Vector Projection