Explain the cross product.

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I love how everyone just knows how to describe the algebra.

i x j does not equal j x i BECAUSE you are multiplying directions (vectors) in respect to each other.

Think of multiplying the front edge of your desk (i) by the side (j). The answer we call k will point downwards through the top of your desk, hence -k.
If we switcheroo and multiply the side (j) by front (i), by mathematical convention, k will point upwards.


Stare at this: https://upload.wikimedia.org/wikipe...vector.svg/484px-Cross_product_vector.svg.png
 
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1. How this calculation yields another vector
2. Why this vector is perpendicular to the other two vectors.
Why is the cross product of the unit vectors i x j=k and j x i= -k?
I think the answer to your questions lies in how the cross product is calculated. It is the determinant of the matrix:
| i j k |
| a b c |
| d e f |

where <a,b,c> and <d,e,f> are vectors in 3-space (toss in a 0 for the z-component, ie for c and/or f, if they are only given as vectors in x and y) and <i,j,k> is a unit vector in R3, ie, i+j+k (where i, j, and k are the components). Compute a few for yourself and you'll start to see what's going on. And also think about the magnitude of the cross produce, |v||w|cos(θ) where v and w are vectors and θ is the angle between them: geometrically, what's going on there?
 
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