Vector Algebra : Repeated addition of a Vector

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Repeated addition of a Vector

what you'll learn...

Overview

Repeated addition of a Vector

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

» Components are multiplied by scalar
→ $\vec{p} = a i + b j + c k$ $vec p = ai + bj + ck$
→ $λ \vec{p} = λ a i + λ b j + λ c k$ $lambda vec p = lambda a i + lambda b j + lambda c k$

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 $\vec{p} + \vec{p}$ $vec p + vec p$ is ' $= 2 \times \vec{p}$ $= 2 times vec p$ '. If $\vec{p} = a i + b j + c k$ $vec p = ai+bj+ck$ then $\vec{p} + \vec{p} = 2 a i + 2 b j + 2 c k$ $vec p + vec p = 2ai+2bj+2ck$ . This equals $2 \times \vec{p}$ $2 times vec p$ .

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

$\vec{p} = a i + b j + c k$ $vec p = ai+bj+ck$
The result of $λ \vec{p}$ $lambda vec p$ is ' $λ a i + λ b j + λ c k$ $lambda a i+ lambda b j + lambda c k$ '.

Can a vector $\vec{p}$ $vec p$ be divided by a real number $λ$ $lambda$ ?
Yes. Division is inverse of multiplication.

If the given vector is $\vec{p} = a i + b j + c k$ $vec p = ai+bj+ck$ then
$\vec{p} \div λ$ $vec p -: lambda$
$= \frac{1}{λ} \times \vec{p}$ $quad quad = 1/lambda xx vec p$
`quad quad = a/lambda i+b/lambda j+c/lambda k

summary

Multiplication of Vector by a scalar: For any vector $\vec{p} = a i + b j + c k \in V$ $vec p = ai + bj+ ck in bbb V$ and scalar $λ \in ℝ$ $lambda in RR$
$λ \vec{p} = λ a i + λ b j + λ c k$ $lambda vec p= lambda ai+ lambda bj + lambda ck$

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

Outline

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