Point multiplication:
In mathematics, the quantity product (dot product; Scalar product, also known as dot product, is a two-dollar operation that accepts two vectors on the real number R and returns a scalar of real values. It is the standard inner product of Euclidean space.
Defined:
value of the point multiplication:The size of u, the size of V, the cosine of the u,v angle. In the premise of u,v nonzero, if the dot product is negative, then the angle formed by U,V is greater than 90 degrees, if it is zero, then the u,v is perpendicular; if positive, the angle of u,v formation is sharp. The dot product of two unit vectors gets the Cos value of the angle of two vectors, through which we can know the similarity of two vectors, and use dot product to determine whether a polygon faces the camera or the camera. The dot product of the vectors is proportional to the cosine of their angle, so in the effect calculation of the spotlight, the illumination effect can be obtained according to the dot product, if the larger the dot product, the smaller the angle, the closer the physical distance from the light, the stronger the illumination.
Operation Law:
Fork Multiplication:The vector product, also called the outer product, the cross product in the mathematics, is called the vector product, the cross multiplication in the physics, is one kind in the vector space the two-dollar operation of the vectors. Unlike dot product, the result of its operation is a vector rather than a scalar. and the cross product of the two vectors is perpendicular to the two vectors.
Properties:Geometrical meaning:
The length of the cross product |axb| can be interpreted as the area of a parallelogram adjacent to A and B.
Mixed product [a b c] = (AXB) c The volume of parallelepiped with a,b,c edges can be obtained.
Algebraic rules:
Anti-Exchange Law:
axb=-bxa
Assignment Law of Addition:
Ax (b+c) =AXB+AXC
Compatible with scalar multiplication:
(RA) Xb=ax (RB) = R (AXB)
Does not satisfy the binding law, but satisfies the Jacobian identity:
Ax (BXC) +bx (CXA) +cx (AXB) =0
The distribution law, linearity and Jacobian identities indicate that R3 with vector addition and cross product constitute a Lie algebra.
Two non-zero vectors A and b parallel, when and only if the axb=0 Lagrange formula
This is a well-known formula and is very useful:
Ax (BXC) =b (a C)-C (a B),
Application:The core of solving the illumination is to find the surface normal of the object, and the cross product operation ensures that the normal can be obtained by the cross product as long as two non-parallel vectors (or three points in the same line) are known on the surface of the object.
Point multiplication and cross multiplication of vectors
