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Vector Cross Product

Vector Cross Product

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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`

Cross Product: Component Form

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.


Keep tapping on the content to continue learning.
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 `vector cross product first principles 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` vector cross product first principles
`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

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Progress

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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.

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