General equation of plane
Ax+By+Cz+D=0, where n=(A, b, c) is the normal vector of the plane, and d is the distance required for the plane to translate to the coordinate origin (so when D=0, the plane passes through the origin).
Modulus (length) of vector
The length of the vector AB (with → on AB) is called the module of the vector, and it is denoted as | AB | (with → on AB) or | A | (with → on A).
Properties of vectors
There are no special rules for the operation of the modulus of a vector. Generally, the modulus of sum and difference of two vectors is calculated by cosine theorem. Orthogonal decomposition method is used to synthesize multiple vectors. If modulus is needed, it is generally necessary to calculate the composite vector first. Modulus is the generalization of absolute value in two-dimensional and three-dimensional space, which can be considered as the length of vector. Extending to high-dimensional space is called norm.
The basic theorem of space vector
* * * Line Vector Theorem Two space vectors A, b (B vector is not equal to 0), a∣∣b is necessary and sufficient if there is a unique real number λ, so that A = λ B.
* * * Vector-oriented Theorem
If the two vectors A and B are not * * * lines, the necessary and sufficient condition for the * * plane of vector C and vectors A and B is that there exists a unique pair of real numbers X and Y, so that c=ax+by.
Space vector decomposition theorem
If the three vectors A, B and C are not * * * planes, there is a unique ordered real array X, Y and Z for any vector P in the space, so that p=xa+yb+zc. Three vectors of any non-* * plane can be used as the basis of space, and the representation of zero vector is unique.
In mathematics, vectors (also known as Euclidean vectors and geometric vectors) refer to quantities with magnitude and direction. It can be imagined as a line segment with an arrow. The arrow indicates the direction of the vector; Line segment length: indicates the size of the vector. The quantity corresponding to a vector is called a quantity (called a scalar in physics), and a quantity (or scalar) has only a size and no direction.
Vector notation: print letters (such as A, B, U, V) in bold, and add a small arrow "→" at the top of the letter when writing. If the starting point A and the ending point B of a vector are given, the vector can be recorded as AB with a → sign at the top. In the space Cartesian coordinate system, vectors can also be expressed in pairs. For example, (2,3) in the xOy plane is a vector.