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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
think,
figure-out, &
learn.

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User Guide

To make best use of nubtrek, understand what is available.

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 Cross Product

Voice

Voice

Home

»  equal vector cross product does not imply vectors are equal

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

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

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

When products of two vectors are equal

plain and simple summary

nub

plain and simple summary

nub

dummy

•  Given cross products are equal, does not imply the vectors are equal.

•  If cross products are equal then difference of the vectors will be collinear to the vector with which cross 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 the property when two vector cross products are equal

Keep tapping on the content to continue learning.
Starting on learning "When products of two vectors are equal". ;;

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

• x=a
• x=b
• x=ab

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 an unknown vec x. Given that vec x xx vec p = vec q xx vec p. What is the value of vec x? Note: All the vectors shown in yellow-dotted-line will form parallelograms of same area with vec p.

• cannot be calculated
• vec x = vec q

The answer is 'Cannot be calculated'.

vec x xx vec p = vec q xx 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 cross product, vec q is split into orthogonal components and the component in perpendicular to vec p is in the product. The component parallel to vec p is lost.

vec x xx vec p = vec q xx vec p imply that the component perpendicular to vec p of both vec x and vec q are equal. If we subtract vec x - vec q then the common component will cancel out and the remaining vector will be parallel to vec p.

vec x xx vec p = vec q xx vec p

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

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

comprehensive information for quick review

Jogger

comprehensive information for quick review

Jogger

dummy

Cannot Cancel: Given vec x xx vec p = vec q xx 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 xx vec p = vec q xx vec p imply that
(vec x - vec q)xx vec p = 0
Which implies vec x - vec q is either vec 0 or is parallel or collinear to vec p.

practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

Progress

Progress

consider real numbers a, b in real numbers, and an unknown real number x. Given that a x = a b, what is x?
1
2
3
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 p, q, in vector space v and an unknown vector x. Given that vector x cross vector p = vector q cross vector p. What is the value of vector x?
cannot;calculated;not
cannot be calculated
vector;x;q
vec x = vec q
The answer is 'Cannot be calculated'.
vector x cross vector p = vector q cross 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 cross product, vector q is split into orthogonal components and the component in perpendicular to vector p is in the product. The component parallel to vector p is lost.
Given cross products are equal, does not imply the vectors are equal.
Cannot Cancel: Given vector x cross vector p = vector q cross vector p, does not imply vector x = vector q. That is the vector p cannot be canceled on both sided of the equation or on numerator and denominator in a division.
Vector x cross vector p = vector q cross vector p, imply that the component perpendicular to vector p of both vector x and vector q are equal. If we subtract vector x minus vector q, then the common component will cancel out and the remaining vector will be parallel to vector p.
The given proof is understood as vector x minus vector q is parallel to vector p.
If cross products are equal then difference of the vectors will be collinear to the vector with which cross products are equal.
Subtraction on sides of an Equation: Vector x cross vector p = vector q cross vector p imply that vector x minus vector q cross vector p = 0. Which implies vector x minus vector q is either vector 0 or is parallel or collinear to vector p.

we are not perfect yet...