Why use cos for dot product and sin for cross product?

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SUMMARY

The discussion clarifies the mathematical reasoning behind using cosine for the dot product and sine for the cross product of vectors. The dot product, defined as A · B = |A| |B| cos(θ), measures the projection of vector A onto vector B, while the cross product results in a vector perpendicular to both A and B, determined by the right-hand rule. Additionally, the cross product represents the area of the parallelogram formed by vectors A and B. This geometric interpretation highlights the complementary nature of these two operations in vector mathematics.

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Dose anybody knw that why we take cos with dot product and Sin with cross product?
 
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Welcome to PF!

They are complementary operations.Tthe dot product gives you the part of vector A projected onto B whereas the cross product gives you the part of A not projected onto B and vice versa.

A dot B = |A| |B| cos(AB) and the project of A on B = |A| cos (AB) = (A dot B) / |B|

The cross product also gives you a vector normal to both A and B using the righthand rule by convention.

There are other geometric ways of looking at it too. The cross product is the area of the parallelogram with A and B as its sides.
 
Last edited:
The dot product is related to the projection of one vector on another. If you draw vectors u and v with "ends" together and drop a perpendicular from the tip of vector u to vector v, then you have a right triangle in which the length of u is the hypotenuse and the length of the projection is the "near side".
 

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