Vector as sum of vectors

Overview

**Vector as Sum of Vectors**

» numerical expression in place of numbers

→ $2$ can be represented with $4-2$ or $\frac{34}{17}$

*All these represent the same number.*

» Vectors as Linear combination of vectors

→ $3\sqrt{2}$ at $45}^{\circ$ angle

→ $3$ at $0}^{\circ$ + $3$ at $90}^{\circ$

→ $6$ at $90}^{\circ$ + $3\sqrt{2}$ at $-{45}^{\circ}$

*All these represent the same vector.*

linear combination

We studied numerical expressions. The following expressions represent the same quantity.

$5$

$2+3$

$6-1$

*A quantity can be equivalently represented with severals forms of numerical expressions. *

Three persons, starting from a common point, are walking

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Person A walks at an angle $45}^{\circ$ for $3\times \sqrt{2}$ meter

•
Person B walks $3$ meter at an angle $0}^{\circ$ and then 3 meter at an angle $90}^{\circ$

•
Person C walks $6$ meter at an angle $90}^{\circ$ and then $3\hspace{0.25em}\times \sqrt{2}$ meter at an angle $-{45}^{\circ}$

To walk back to the starting point, all three has to walk 'same distance in the same direction'.

The three vectors given above are effectively the same vector in three different forms. *A vector quantity can be equivalently represented by combination of several vectors. *

The walk of Person A (at an angle $45}^{\circ$ for $3\times \sqrt{2}$ meter) is one vector quantity $3\sqrt{2}$ at angle $45}^{\circ$

The walk of Person B ()$3$ meter at an angle $0}^{\circ$ and then 3 meter at an angle $90}^{\circ$) is sum of two vector quantities $3$ at $0}^{\circ$ and $3$ at $90}^{\circ$.

Note that person A’s and person B’s walks are specified by one vector quantity and two vector quantities. Still, they are at the same distance and direction from the starting point. Effectively, their return to the starting point can be specified by one identical vector quantity.

To understand this, Consider the example in scalar quantities. Person A has $3$ apples in a basket. Person B has $2$ apples in one hand and $1$ apple in another hand. Both has the same number of apples, which is understood as $2+1\hspace{0.25em}=\hspace{0.25em}3$.

Given vectors

$\overrightarrow{x}=2i+3j+4k$

$\overrightarrow{y}=-1i-1j-2k$

$\overrightarrow{z}=i+2j+2k$

Note that $\overrightarrow{x}+\overrightarrow{y}=i+2j+2k$ is same as $\overrightarrow{z}$. So $\overrightarrow{z}$ can be equivalently given as $\overrightarrow{x}+\overrightarrow{y}$. Reiterating that *A vector can be equivalently given as a sum of vectors.*

summary

**Linear Combination of Vectors: **A vector can be equivalently represented as sum of vectors.*A vector can be equivalently given as the sum of vectors.*

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__