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Vector algebra and spatial analytic geometry
This is a note on vector algebra and spatial analytic geometry, covering vectors and their linear operations, quantity products, vector products, mixed products, planes and their equations, space straight lines and their equations.
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Vector algebra and spatial analytic geometry
- bacteria
This is a mind map about bacteria, and its main contents include: overview, morphology, types, structure, reproduction, distribution, application, and expansion. The summary is comprehensive and meticulous, suitable as review materials.
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- Outline
Chapter 8 (Vector Algebra and Space Analytic Geometry)
Section 1 Vectors and their linear operations
vector concept
Only the magnitude and direction of the vector are considered, regardless of the starting point
The size of the vector is called the module of the vector: | r | = √(x2 y2 z2). A vector with modulus equal to 1 is called a unit vector
Linear operations on vectors
Addition and subtraction of vectors: triangle rule, parallelogram rule
Commutative law a b=b a
Associative law (a b) c=a (b c)
Multiplication of vectors and numbers
Associative law λ(μa)=μ(λa)=(λμ)a
Distributive law (λ μ)a=λa μa λ(a b)=λa λb
Theorem 1 Assume vector a≠0, then the necessary and sufficient condition for vector b to be parallel to a is: there is a unique real number λ such that b=λa.
space rectangular coordinate system
Hanging limit: Above the xOy plane, in front of the yOz plane, and to the right of the zOx plane is the first hanging limit (Ⅰ), and the rest are arranged clockwise.
Vector r=xi yj zk.
Using coordinates to perform linear operations on vectors
Modulus, direction angle, and projection of vectors
The angles α, β, and γ between the non-zero vector r and the three coordinate axes are called the direction angles of the vector r.
Section 2 Quantity product, vector product, mixed product
quantity product
Commutative law a·b=b·a
Distributive law (a b)·c=a·c b·c
Associative law (λa)·b=λ(a·b), λ is a number
vector product
b x a=-a x b
Distributive law (a b) x c=a x c b x c
Associative law (λa) x b=a x (λb)=λ(a x b), (λ is a number)
The necessary and sufficient condition for vector a//b is a x b=0
Section 3 Plane and its equations
French style
general equation of plane
The intercept equation of the plane, a, b, c are the intercepts of the plane on the x, y, z axis in turn.
The angle between the normal vectors of two planes (usually an acute angle or a right angle) is called the angle between the two planes
Section 4 Space straight lines and their equations
general equations of straight lines in space
Symmetry equation or point equation of a straight line: A straight line can be drawn through a point in space and can only be drawn parallel to a known straight line, so when there is a point L on the straight line, there is a point M. When (x., y., z.) and its direction vector s=(m,n,p) are known, L can be found.
Parametric equation of a straight line (can be derived from the symmetry equation)
The angle between the direction vectors of two straight lines (usually an acute angle or a right angle) is called the angle between the two straight lines.
When the straight line is not perpendicular to the plane, the angle φ (0≤φ<π/2) between the straight line and its projected straight line on the plane is called the angle between the straight line and the plane.
Section 5 Surfaces and Their Equations
There is a known curve C on the yOz coordinate surface, and the equation is f(y,z)=0
Section 6 Space Curves and Their Equations
General equation of space curve C
Parametric equations of space curves
For example, the parametric equation of a spiral
The cylinder must contain curve C. The cylinder with curve C as the alignment and the generatrix parallel to the z-axis (that is, perpendicular to the x0y plane) is called the projection cylinder of curve C on the x0y plane. The intersection of the projected cylinder and the x0y plane is called space. The projection curve of curve C on the x0y plane, or simply the projection
The surface represented must contain the projection of the space curve on the xOy surface