- #1
okkvlt
- 53
- 0
Say i want to find [xa,ya,za]*[xb,yb,zb]
I could use pythagoras to find |a| and |b|. Then use pythagoras again to find the distance between the endpoints of a and b. Then use law of cosines to find the angle formed by a and b, then do |a||b|cos(angle).
Or, i could just do xa*xb+ya*yb+za*zb.
How are all these operations compressed into such a simple form?
(By the way, the dot product in 2 dimensions is the area of the parallelogram formed by the endpoints of a+b, a, b, and 0, right?) Then the dot product of vectors in 3 dimensions is the area of the parallelogram squared to get a 3 dimensional shape, right? Whats the signifigance of a negative dot product?
And if i want to find the cross product, all i have to do is arrange them like this
xa you za
xb yb zb
And for each coordinate of the cross product i just remove that column and find the determinant of the remaining 2x2 matrix, reversing the sign for y. Doing it otherwise, i would have to do a lot of complicated things, especially finding that perpendicular unit vector.
Also, how do matrix determinants work in finding the solution to systems of equations?
How do matrices and vectors work?
I could use pythagoras to find |a| and |b|. Then use pythagoras again to find the distance between the endpoints of a and b. Then use law of cosines to find the angle formed by a and b, then do |a||b|cos(angle).
Or, i could just do xa*xb+ya*yb+za*zb.
How are all these operations compressed into such a simple form?
(By the way, the dot product in 2 dimensions is the area of the parallelogram formed by the endpoints of a+b, a, b, and 0, right?) Then the dot product of vectors in 3 dimensions is the area of the parallelogram squared to get a 3 dimensional shape, right? Whats the signifigance of a negative dot product?
And if i want to find the cross product, all i have to do is arrange them like this
xa you za
xb yb zb
And for each coordinate of the cross product i just remove that column and find the determinant of the remaining 2x2 matrix, reversing the sign for y. Doing it otherwise, i would have to do a lot of complicated things, especially finding that perpendicular unit vector.
Also, how do matrix determinants work in finding the solution to systems of equations?
How do matrices and vectors work?