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Matrix Size & Validation
Vector Algebra
What is Vector?
Vector Norm
Unit Vector
Vector Addition
Vector Subtraction
Vector Scalar Multiple
Vector Multiplication
Vector Inner Product
Vector Outer Product
Vector Cross Product
Vector Triple Cross Product
Vector Triple Dot Product
Scalar Triple Product
Orthogonal & Orthonormal Vector
Cos Angle of Vectors
Scalar and Vector Projection
Matrix Algebra
What is a matrix?
Special Matrices
Matrix One
Null Matrix
Matrix Diagonal Is Diagonal Matrix?
Identity Matrix
Matrix Determinant
Matrix Sum
Matrix Trace
Matrix Basic Operation
Is Equal Matrix?
Matrix Transpose
Matrix Addition
Matrix Subtraction
Matrix Multiplication
Matrix Scalar Multiple
Hadamard Product
Horizontal Concatenation
Vertical Concatenation
Elementary Row Operations
Matrix RREF
Finding inverse using RREF (Gauss-Jordan)
Finding Matrix Rank using RREF
Matrix Inverse
Is Singular Matrix?
Linear Transformation
Matrix Generalized Inverse
Solving System of Linear Equations
Linear combination, Span & Basis Vector
Linearly Dependent & Linearly Independent
Change of basis
Matrix Rank
Matrix Range
Matrix Nullity & Null Space
Eigen System
Matrix Eigen Value & Eigen Vector
Symmetric Matrix
Matrix Eigen Value & Eigen Vector for Symmetric Matrix
Similarity Transformation and Matrix Diagonalization
Matrix Power
Orthogonal Matrix
Spectral Decomposition
Singular Value Decomposition
Resources on Linear Algebra

Vector Projection

By Kardi Teknomo, PhD.

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Suppose we have a vector Vector Projectionin 2-dimensional space, and we would like to find the component of vector Vector Projection in the direction of horizontal and vertical axis. Using trigonometry we can find the magnitude of vector component to be Vector Projection andVector Projection. The direction of these vector components are the standard unit vector of horizontal and vertical axes Vector Projection andVector Projection.
Vector Projection

We can find vector projection of a vector Vector Projectiononto other vector Vector Projectionbased on the inner product of the two vectors. Scalar projection of vector Vector Projectiononto vector Vector Projection is the magnitude of projection given asVector Projection. Vector projection of vector Vector Projectiononto vector Vector Projection is the magnitude of projection times the unit vector of Vector ProjectionbecomesVector Projection. Geometrically, vector projection is shown in the figure below.
Vector Projection

Projection of vector Vector Projectiononto horizontal axis is equivalent to projection of vector Vector Projectiononto unit standard vectorVector Projection .  The scalar projection isVector Projection. The projection vector becomesVector Projection. Cosine angle between two vectors is Vector Projection. Thus, Vector Projection.

The interactive program below will give you the projection vectors and the scalar projection. You need to provide 2 vectors as the input. Random Example button will generate random input vectors.

vector x vector y

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Preferable reference for this tutorial is

Teknomo, Kardi (2011) Linear Algebra tutorial. http:\\people.revoledu.com\kardi\ tutorial\LinearAlgebra\


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