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Thought-Process to Discover Knowledge

Welcome to **nub****trek**.

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.

Read in the blogs more about the unique learning experience at nubtrek.continue

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.

Read in the blogs more about the unique learning experience at nubtrek.continue

Welcome to **nub****trek**.

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The content is presented in small-focused learning units to enable you to

think,

figure-out, &

learn.

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.

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trek,

jogger,

exercise.

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» 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`

*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.

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

*your progress details*

Progress

*About you*

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.