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Thought-Process to Discover Knowledge

Welcome to nubtrek.

Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

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User Guide

Welcome to nubtrek.

The content is presented in small-focused learning units to enable you to
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figure-out, &
learn.

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User Guide

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nubtrek is designed to explain mathematics and science for young readers. Every topic consists of four sections.

nub,

trek,

jogger,

exercise.

User Guide

nub is the simple explanation of the concept.

This captures the small-core of concept in simple-plain English. The objective is to make the learner to think about.

User Guide

trek is the step by step exploration of the concept.

Trekking is bit hard, requiring one to sweat and exert. The benefits of taking the steps are awesome. In the trek, concepts are explained with exploratory questions and your thinking process is honed step by step.

User Guide

jogger provides the complete mathematical definition of the concepts.

This captures the essence of learning and helps one to review at a later point. The reference is available in pdf document too. This is designed to be viewed in a smart-phone screen.

User Guide

exercise provides practice problems to become fluent in the concepts.

This part does not have much content as of now. Over time, when resources are available, this section will have curated and exam-prep focused questions to test your knowledge.

summary of this topic

### Properties of Dot Product

Voice

Voice

Home

»  equal vector dot product does not imply vectors are equal

→  vec p cdot vec q   = vec p cdot vec r  does not imply that vec q = vec r

→  on both sides of the equation, vec p cannot be canceled.

→  vec p cdot vec q   = vec p cdot vec r  imply that vec p cdot (vec q - vec r) = 0

### When products of two vectors are equal

plain and simple summary

nub

plain and simple summary

nub

dummy

•  Equal dot products does not imply the vectors are equal.

•  If dot products are equal then difference of the vectors will be perpendicular to the vector with which dot products are equal.

simple steps to build the foundation

trek

simple steps to build the foundation

trek

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In this page, you will learn whether vectors can be canceled in both side of the equation when two vector dot products are equal.

Keep tapping on the content to continue learning.
Starting on learning conditions in which "vector dot products of two vectors are equal". ;; In this page, you will learn whether vectors can be canceled in both side of the equation when two vector dot products are equal.

Consider real numbers a, b in RR and an unknown number x. Given that ax = ab, what is the value of x?

• x=a
• x=b
• x = a b

The answer is 'x=b'. Both the left hand side and right hand side of the equation is divided by a to arrive at the solution.

Consider vectors vec p, vec q in bbb V and a unknown vec x. Given that vec x cdot vec p = vec q cdot vec p. What is the value of vec x? Note: All the vectors shown in yellow-dotted-line will have the same projection on to the vec p.

• cannot be calculated
• vec x = vec q

The answer is 'Cannot be calculated'.

vec x cdot vec p = vec q cdot vec p does not imply that vec x = vec q.
That is, vec p cannot be canceled on left-hand-side and right-hand-side. Note that in dot product, vec q is split into orthogonal components and the component in parallel to vec p is only in the product. The component perpendicular to vec p is lost.

vec x cdot vec p = vec q cdot vec p imply that both vec x and vec q has same projection on to vec p shown as a If we subtract vec x - vec q then the common component will cancel out and the remaining vector will be perpendicular to vec p.

vec x cdot vec p = vec q cdot vec p

Subtracting vec q cdot vec p from both the sides. vec x cdot vec p - vec q cdot vec p = vec q cdot vec p - vec q cdot vec p
(vec x - vec q)cdot vec p = (vec q - vec q)cdot vec p
(vec x - vec q)cdot vec p = 0 cdot vec p
(vec x - vec q)cdot vec p = 0

The above can be understood as vector vec x - vec q is orthogonal to vec p.

comprehensive information for quick review

Jogger

comprehensive information for quick review

Jogger

dummy

Cannot Cancel: Given vec x cdot vec p = vec q cdot vec p does not imply vec x = vec q. That is, the vec p cannot be canceled on both sides of the equation or on numerator and denominator in a division.

Subtraction on sides of an Equation: vec x cdot vec p = vec q cdot vec p imply that
(vec x - vec q)cdot vec p = 0
Which implies vec x - vec q is either vec 0 or is perpendicular to vec p.

practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

Progress

Progress

considers a,b in real numbers, and an unknown number x. Given that a x = a b, what is the value of x?
1
2
3
The answer is "x equals b". Both the left hand side and right hand side of the equation is divided by a to arrive at the solution.
consider vectors p, q and an unknown vector x. Given that vector x dot vector p = vector q dot vector p. What is the value of vector x?
1
2
The answer is 'Cannot be calculated'.
vector x dot vector p = vector q dot vector p does not imply that vector x = vector q. That is vector p cannot be canceled on left-hand-side and right-hand-side. ;; note that in dot product, vector q is split into orthogonal components and the component in parallel to vector p is only in the product. The component perpendicular to vector p is lost.
Equal dot products does not imply the vectors are equal.
Cannot Cancel: Given vector x dot vector p = vector q dot vector p ;; does not imply vector x = vector q. This is, vector p cannot be canceled on both sides of the equation or on numerator and denominator in a division.
vector x dot vector p = vector q dot vector p imply that both vector x and vector q has same projection on to vector p, shown as a. If we subtract vector x minus vector q, then the common component will cancel out and the remainint vector is perpendicular to vector p.
Simple manipulation of an equation involving two dot products is shown. The result that vector x - vector q is orthogonal to vector p is derived.
If dot products are equal then difference of the vectors will be perpendicular to the vector with which dot products are equal.
Subtraction on sides of an Equation: vector x dot vector p = vector q dot vector p imply that ;; vector x minus vector q, dot vector p = 0. Which implies vector x minus vector q is either vector 0 or is perpendicular to vector p.

we are not perfect yet...