maths > vector-algebra

Repeated addition of a Vector

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Repeated addition of a Vector

 »  Repeated addition is generalized to Multiplication of a vector by Scalar

 »  Components are multiplied by scalar
    →  p=ai+bj+ck
    →  λp=λai+λbj+λck

repeated addition

vector repeated addition

We have learned about addition of two or more vectors. Consider a vector that is added to itself. The result of the addition p+p is ' =2×p'. If p=ai+bj+ck then p+p=2ai+2bj+2ck. This equals 2×p.

multiplication of vector by scalar

Repeated addition can be generalized to multiplication of vector by a scalar. Scalar multiplier is denoted with λ to identify differently to the component values a,b,c. Note that the scalar multiplier and component values are real numbers.

The result of λp is 'λai+λbj+λck'.

Can a vector p be divided by a real number λ?
Yes. Division is inverse of multiplication.

If the given vector is p=ai+bj+ck then
`quad quad = a/lambda i+b/lambda j+c/lambda k


Multiplication of Vector by a scalar: For any vector p=ai+bj+ckV and scalar λ

When vector is multiplied by a scalar, the vector scales up or down proportionally.


The outline of material to learn vector-algebra is as follows.

Note: Click here for detailed outline of vector-algebra.

•   Introduction to Vectors

    →   Introducing Vectors

    →   Representation of Vectors

•   Basic Properties of Vectors

    →   Magnitude of Vectors

    →   Types of Vectors

    →   Properties of Magnitude

•   Vectors & Coordinate Geometry

    →   Vectors & Coordinate Geometry

    →   Position Vector of a point

    →   Directional Cosine

•   Role of Direction in Vector Arithmetics

    →   Vector Arithmetics

    →   Understanding Direction of Vectors

•   Vector Addition

    →   Vector Additin : First Principles

    →   Vector Addition : Component Form

    →   Triangular Law

    →   Parallelogram Law

•   Multiplication of Vector by Scalar

    →   Scalar Multiplication

    →   Standard Unit Vectors

    →   Vector as Sum of Vectors

    →   Vector Component Form

•   Vector Dot Product

    →   Introduction to Vector Multiplication

    →   Cause-Effect-Relation

    →   Dot Product : First Principles

    →   Dot Product : Projection Form

    →   Dot Product : Component Form

    →   Dot Product With Direction

•   Vector Cross Product

    →   Vector Multiplication : Cross Product

    →   Cross Product : First Principles

    →   Cross Product : Area of Parallelogram

    →   Cross Product : Component Form

    →   Cross Product : Direction Removed