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mathsVector AlgebraVector Cross Product

Cross Product: Component Form

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



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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`

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

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) |`

                            
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