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mathnerd15
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how do you prove the distributive quality of the cross and dot products?
thanks very much!
thanks very much!
mathnerd15 said:how would you derive the concept of geometry from first principles?
Is that your way of saying that the answer you got is inadequate? It is not. What he's suggesting is just that you use the definitions of the dot product and the cross product to rewrite both the left-hand side and the right-hand side of the equality you want to prove. Then you just compare the results. Are they equal?mathnerd15 said:how would you derive the concept of geometry from first principles?
mathnerd15 said:|A||B+C|costheta3=|A|B|costheta1+|A||C|costheta2
|A||B+C|sintheta3n=|A||B|sintheta1n+|A||C|sintheta2n
I was referring to a different problem... I mean Valenzia refers to deriving geometry from general concepts and it was fascinating to see an exposition of that
mathnerd15 said:thanks!
but isn't Euclid land far away from modern mathematics?
I was thinking that you should use the definition $$A\cdot B=\sum_{i=1}^3 A_i B_i$$ of the dot product, and a similar definition of the cross product. This seems easier to me than to use the formulas ##A\cdot B=|A||B|\cos\theta## and ##|A\times B|=|A||B|\sin\theta##. If you want to continue with the approach you have started, you will have to figure out how the angles you called ##\theta_1,\theta_2,\theta_3## are related. In the case of the cross product, you have the additional problem that it's not sufficient to prove that ##\left|A\times(B+C)\right| =\left|A\times B+A\times C\right|##. You also need to verify that ##A\times(B+C)## is in the same direction as ##A\times B+A\times C##.mathnerd15 said:|A||B+C|costheta3=|A|B|costheta1+|A||C|costheta2
|A||B+C|sintheta3n=|A||B|sintheta1n+|A||C|sintheta2n
If you use that definition to rewrite ##A\times (B+C)## and ##A\times B+A\times C##, you can easily see that the two are equal.mathnerd15 said:as you stated in component form, AXB=(AyBz-AzBy)xhat...it seems worth memorizing since the matrix notation is a bit hard to read...
More general than what? If you use the definition (any version of it), you can prove that for all ##A,B,C\in\mathbb R^3##, we have ##A\times (B+C)=A\times B + A\times C##.mathnerd15 said:there's supposed to be a more general proof
http://en.wikipedia.org/wiki/Cross_product
If you click on the quote button next to one of my posts, you can see how I did it. If you want to know more, check out this FAQ post about it.mathnerd15 said:by the way how do I use the nice latex notation?
I had a quick look at these proofs. They prove the same statements that we're discussing here.mathnerd15 said:in G.E. Hays Vector and Tensor Analysis there are proofs of the general case, chapter 1, section 7 and 8
The formula for calculating the cross product of two vectors, u and v, is given by the determinant:
u x v = |i j k|
|u1 u2 u3|
|v1 v2 v3|
where i, j, and k are unit vectors in the x, y, and z directions, and u1, u2, u3 and v1, v2, v3 are the components of the two vectors in each direction.
The cross product of two vectors gives a new vector that is perpendicular to both of the original vectors. The magnitude of the cross product is equal to the area of the parallelogram formed by the two vectors, and the direction of the cross product follows the right-hand rule.
Yes, the cross product of two vectors can be zero. This happens when the two vectors are parallel or anti-parallel to each other. In this case, the area of the parallelogram formed by the two vectors is zero, and the resulting vector is also zero.
The dot product and cross product are two different ways of multiplying two vectors. The dot product gives a scalar quantity, while the cross product gives a vector quantity. The dot product can help determine the angle between two vectors, while the cross product can help determine the direction and magnitude of a perpendicular vector.
The cross product has various applications in physics, engineering, and graphics. Some examples include calculating the torque exerted by a force on a lever, finding the angular momentum of a rotating object, and determining the direction of the magnetic field around a current-carrying wire. In graphics, the cross product is used to calculate lighting and shading effects in 3D computer graphics.