In Vector Multiplications why Dot product uses cosine and Cross product uses sin ?
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Dot product geometrically:
1. Take vectors A and B.
2. Do scalar projection of A onto B. In other words take the length of the component of A that is in same direction as B (negative if opposite direction).
3. Multiply the length of B with the length of the scalar projection.
Picture: https://en.wikipedia.org/wiki/File:Dot_Product.svg
If you look at the triangles this creates you’ll see that the length of the scalar projection of A onto B is |A|cos(θ) where θ is the angle between the two vectors.
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Cross product geometrically:
1. Take vectors A and B.
2. Take the surface area of the parallelogram that is fromed from the vectors A and B. This surface area will be the length of the cross product.
3. Set the cross product vector to be orthogonal to both A and B according to the right hand rule (image https://en.wikipedia.org/wiki/File:Right_hand_rule_cross_product.svg).
Picture: https://en.wikipedia.org/wiki/File:Cross_product_parallelogram.svg
If you again look at the triangles that form the parallelogram you will see that its surface area is |A||B||sin(θ)| where θ is again angle between A and B. (to see it easier split the parallelogram into two triangles of same size with sides formed by A and B).
You can’t really explain vector products to a 5 year old, but here’s the best I can do without pictures.
The sine and cosine functions describe the motion of a point rotating around the origin counterclockwise, with respect to the angle to the x axis. Cosine gives the x coordinate and sine gives the y coordinate.
Take your two vectors, and move and rotate the picture so that they start at the origin and one of them points to the right along the x axis.
The dot product is the position of one vector projected perpendicularly onto the other, times the length of the other vector. But projecting the non-x-axis vector onto the x axis is the same as just taking its x coordinate. Since cosine tracks the x coordinate, there’s a cosine in the formula.
The cross product is the area of the parallelogram formed by the vectors. Parallelogram area is base times height. But the height is just the y coordinate of the top of the non-x-axis vector, which is given by its length times the sine of the angle.
Because that is how we have chosen to define them.
We could define a third kind of vector product, say an @ product, where:
> **a**@**b** = |**a**|.|**b**|.sin(θ)
or a cos version of the cross product. And those would be perfectly valid and sensible things. Except it turns out things like this aren’t quite as useful or mathematically interesting.
A lot of our vector mathematics was developed by physicists, particularly those working on electromagnetism, and it turns out that the dot and cross products are really useful in a lot of physics (the dot product relating to resolving vectors in the same direction as each other, the cross product for finding things in perpendicular directions or for doing circular motion).
Once we have got these two definitions they can be generalised more broadly into the “exterior” (or “wedge”) and “inner” products – which take the same sort of idea but apply them to more generalised vector spaces.
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