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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**.

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

think,

figure-out, &

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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.

nub,

trek,

jogger,

exercise.

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Vector Cross Product: Component Form

`vec p xx vec q = |(i, j, k),(p_x, p_y, p_z),(q_x, q_y, q_z) |``=(p_yq_z-p_zq_y)i` `+ (p_zq_x-p_xq_z)j` `+ (p_xq_y-p_yq_x)k`

*plain and simple summary*

nub

*plain and simple summary*

nub

dummy

Vector Cross Product between two vectors in component form is defined as a vector from a determinant.

*simple steps to build the foundation*

trek

*simple steps to build the foundation*

trek

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In this page, you will learn about the component form of the vector cross product.

Starting on learning "component form of cross product". ;; In this page, you will learn about the component form of the vector cross product.

Given `vec p = p_x i + p_yj+p_zk` and `vec q =q_x i + q_yj+q_zk ` How to find `sin theta`, or the cross product in the component form?

Component form of vector cross product : geometrical proof

`|vec p xx vec q|`

`quad quad = |p||q|sin theta`

`quad quad = |p||q|sqrt(1-cos^2 theta)`

`quad quad = sqrt(|p|^2|q|^2-|p|^2|q|^2cos^2 theta)`

`quad quad = sqrt(|p|^2|q|^2-|p cdot q|^2)`

Substitute the following

`|vec p|^2 = (p_x^2+p_y^2+p_z^2)`

`|vec q|^2 = (q_x^2+q_y^2+q_z^2)`

`|vec p cdot vec q|^2 = (p_xq_x+p_yq_y+p_zq_z)^2`

and after some algebraic rearrangement, the following is arrived at

`|vec p xx vec q|`

`quad quad = sqrt((p_yq_z-p_zq_y)^2)`

`quad quad quad quad bar(+(p_zq_x-p_xq_z)^2) `

`quad quad quad quad bar(+(p_xq_y-p_yq_x)^2)`

`quad quad = | (p_yq_z-p_zq_y)i`

`quad quad quad quad + (p_zq_x-p_xq_z)j`

`quad quad quad quad + (p_xq_y-p_yq_x)k |`

It is proven that

`vec p xx vec q`

`quad quad = (p_yq_z-p_zq_y)i`

`quad quad quad quad + (p_zq_x-p_xq_z)j`

`quad quad quad quad + (p_xq_y-p_yq_x)k`

The same can be written in the determinant form

`vec p xx vec q = |(i, j, k),(p_x, p_y, p_z),(q_x, q_y, q_z)|`

*Bilinear Property: * For any vector `vec p, vec q, vec r in bbb V` and `lambda in RR`

`(lambda vec p + vec q) xx vec r ``= lambda (vec p xx vec r) + (vec q xx vec r)`

This is explained and proven in properties of the cross product. For now, consider this to be true.

A vector `vec p = p_x i + p_y j + p_z k` is sum of scalar multiple of vectors. `i, j, k` are unit vectors, and the scalar multiples are `p_x, p_y, p_z`.

The same applies to `vec q = q_x i + q_y j + q_z k` -- sum of multiple of vectors.

Proof for component form of vector cross product using bilinear property of cross product.

`vec p xx vec q`

`quad quad = (p_x i + p_y j + p_z k) xx`

`quad quad quad quad (q_x i + q_y j + q_z k)`

Applying bilinear property of cross product

`quad quad = p_x i xx (q_x i + q_y j + q_z k) `

`quad quad quad quad + p_y j xx (q_x i + q_y j + q_z k)`

`quad quad quad quad + p_z k xx (q_x i + q_y j + q_z k)`

Applying `i xx i = 0`, `j xx j = 0`, `k xx k = 0`

`i xx j = k`, `j xx k = i`, `k xx i = j`

`j xx i = -k`, `k xx j = -i`, `i xx k = -j`

After algebraic rearrangement, we get

`vec p xx vec q`

`quad quad = (p_yq_z-p_zq_y)i`

`quad quad quad quad + (p_zq_x-p_xq_z)j`

`quad quad quad quad + (p_xq_y-p_yq_x)k`

*comprehensive information for quick review*

Jogger

*comprehensive information for quick review*

Jogger

dummy

**Vector Cross Product: ** for vectors `vec p, vec q in RR^3`

`vec p xx vec q = |(i, j, k),(p_x, p_y, p_z),(q_x, q_y, q_z) |`

*practice questions to master the knowledge*

Exercise

*practice questions to master the knowledge*

Exercise

*your progress details*

Progress

*About you*

Progress

Given vector p = p x i + p y j + p z k and vector q = q x i + q y j + q z k. How to find sine theta , or the cross product in the component form?

Component form of vector cross product : geometrical proof is given.

The vector cross product can be written in the determinant form.

Bilinear Property: For any vector p, q, r in vector space v and lambda in RR ;; lambda vector p + vector q, cross vector r = lambda, vector p cross vector r, + vector q cross vector r;; This is explained and proven in properties of the cross product. For now, consider this to be true.

A vector p = p x i + p y j + p z k is sum of scalar multiple of vectors. i, j, k are unit vectors. and the scalar multiples are p x, p y, p z. The same applies to vector q - sum of multiple of vectors.

Proof for component form of vector cross product using bilinear property of cross product ; is given.

Vector Cross Product between two vectors in component form is defined as a vector from a determinant.

Vector Cross Product: for vectors p and q in real number 3D space. Vector P cross vector q = determinant, i , j, k ;; p x, p y, p z ;; q z, q y, q z.