Truth proposition: a statement judged to be true.
False proposition: a statement that is judged to be wrong.
2. The condition called a proposition in the form of "if, then" is called the conclusion of the proposition.
3. For two propositions, if the conditions and conclusions of one proposition are the conclusions and conditions of the other proposition, it is called reciprocal proposition. One proposition is called the original proposition and the other is called the inverse of the original proposition.
If the original proposition is "if, then", its inverse proposition is "if, then".
4. For two propositions, if the conditions and conclusions of one proposition happen to be the negation of the conditions and conclusions of the other proposition, then these two propositions are called mutually negative propositions. One of them is called the original proposition and the other is called the negative proposition of the original proposition.
If the original proposition is "if, then", then its negative proposition is "if, then".
5. For two propositions, if the conditions and conclusions of one proposition happen to be the negation of the conclusions and conditions of the other proposition, then these two propositions are called mutually negative propositions. One of them is called the original proposition and the other is called the negative proposition of the original proposition.
If the original proposition is "if, then", then its negative proposition is "if, then".
6, the authenticity of the four propositions:
Primitive proposition
contrary/opposite
No proposal
Reverse negative proposition
real
real
real
real
real
wrong
wrong
real
wrong
real
real
real
wrong
wrong
wrong
wrong
The relationship between truth and falsehood of four propositions;
These two propositions are mutually negative, and they have the same truth;
Two propositions are contradictory propositions or contradictory propositions, and their truth is irrelevant.
7. If, it is a sufficient condition and a necessary condition.
If so, it is a necessary and sufficient condition (sufficient and necessary condition).
8. Connect the propositions with the conjunction "and" to get a new proposition, which is recorded as.
When all are true propositions, they are true propositions; When one of the two propositions is false, it is false.
Connect the propositions with the conjunction "or" and get a new proposition, which is recorded as.
When one of the two propositions is true, it is true; When both propositions are false, it is a false proposition.
Completely negate a proposition and get a new proposition, which is recorded as.
If it is a true proposition, it must be a false proposition; If it is a false proposition, it must be a true proposition.
9. The phrases "to all" and "to any one" are usually called full-name quantifiers in logic, which are represented by "".
A proposition that contains a full-name quantifier is called a universal proposition.
The full name proposition "Any one of them is true" is recorded as ",".
The phrases "you yi" and "at least one" are usually called existential quantifiers in logic, which are represented by "".
Propositions containing existential quantifiers are called special propositions.
The proposition "Being One and Making It True" is specially called ",".
10, full name proposition:, its negation:, the negation of full name proposition is a special proposition.
1 1, the locus of a point whose sum of the distances from two fixed points on a plane is equal to a constant (greater than) is called an ellipse. These two fixed points are called the focal points of the ellipse, and the distance between the two focal points is called the focal length of the ellipse.
12, geometric properties of ellipse:
Focus position
The focus is on the axis
The focus is on the axis
chart
Standard equation
range
and
and
pinnacle
,
,
,
,
axial length
The length of the minor axis and the length of the major axis.
concentrate
,
,
focal distance
symmetrical
Symmetry of Axis, Axis and Origin
weird
collinearity equation
13, let it be any point on the ellipse, the distance from the point to the corresponding directrix is, and the distance from the point to the corresponding directrix is.
14, the locus of the point whose absolute value is equal to a constant (less than) is called hyperbola. These two fixed points are called focal points of hyperbola, and the distance between the two focal points is called focal length of hyperbola.
15, geometric properties of hyperbola:
Focus position
The focus is on the axis
The focus is on the axis
chart
Standard equation
range
Or,
Or,
pinnacle
,
,
axial length
Imaginary axis length and real axis length.
concentrate
,
,
focal distance
symmetrical
About the axis, axis symmetry, center symmetry about the origin
weird
collinearity equation
Asymptote equation
16, the hyperbola with equal length of real axis and imaginary axis is called equilateral hyperbola.
17, let it be any point on the hyperbola, the distance from the point to the corresponding directrix is, and the distance from the point to the corresponding directrix is.
18, the locus of a point on the plane with the same distance from a fixed point and a fixed line is called parabola. The fixed point is called the focus of parabola, and the fixed line is called the directrix of parabola.
19, the focus of the parabola is a line segment perpendicular to the symmetry axis and intersecting with the parabola at two points, which is called the "trajectory" of the parabola.
20, focus radius formula:
If the point is on a parabola and the focus is on, then;
If the point is on a parabola and the focus is on, then;
If the point is on a parabola and the focus is on, then;
If the point is on a parabola and the focus is on, then.
2 1, geometric properties of parabola:
Standard equation
chart
pinnacle
axis of symmetry
axis
axis
concentrate
collinearity equation
weird
range
22, the concept of space vector:
In space, quantities with magnitude and direction are called space vectors.
A vector can be represented by a directed line segment. The length of the directed line segment indicates the size of the vector, and the direction pointed by the arrow indicates the direction of the vector.
The size of the vector is called the modulus (or length) of the vector, and it is recorded as.
A vector with a modulus (or length) is called a zero vector; Modulated vectors are called unit vectors.
Vectors of equal length and opposite directions are called inverse quantities.
Vectors with the same direction and equal modulus are called equal vectors.
23, space vector addition and subtraction:
The operation of finding the sum of two vectors is called vector addition and follows the parallelogram rule. That is, two known vectors with the same starting point in space are parallelograms, and the diagonal of the starting point is. This method of finding vector sum is called parallelogram rule of vector addition.
The operation of finding the difference between two vectors is called vector subtraction, which follows the triangle rule, that is, take any point in the space, make, and then.
24. The product of a real number and a space vector is a vector, which is called vector multiplication. It was the same direction at that time; At this time, in the opposite direction; At that time, it was a zero vector, and the length was recorded as. Twice the length.
25. Let, be a real number and be any two vectors in the space, then the number multiplication operation satisfies the distribution law and the associative law.
Distribution law:; Association rule:
26. If the lines representing the directed line segments of space are parallel or coincident with each other, these vectors are called * * * line vectors or parallel vectors, and zero vectors and arbitrary vectors are defined as * * * lines.
27. Necessary and sufficient condition of vector * * * line: For any two vectors in space, the necessary and sufficient condition is the existence of real numbers, so that.
28. A vector parallel to the same plane is called a * * * vector.
29. Vector * * * surface theorem: The necessary and sufficient condition for a point in space to be in the plane is that there are ordered real number pairs, so; Or at any point in space; Or if it is four o'clock, * * *, then.
30. Given the sum of two non-zero vectors, take any point in the space and make,,, which is called the included angle of the vectors and is recorded as. The included angle range of the two vectors is:.
3 1, for the sum of two non-zero vectors, if, then the vectors are perpendicular to each other, and recorded as.
32, known as the sum of two non-zero vectors, called the quantitative product, recorded as. That is, the quantitative product of. Zero vector and arbitrary vector are both.
33 equals the product of the length of and the projection in the direction.
34, if it is a non-zero vector and a unit vector, then there is;
; ,,;
; .
35. Algorithm of vector number product:;
.
36. If,, are three orthogonal vectors in space, there is an ordered real array for any vector in space, so,, is a component of the upper vector.
37. Basic theorem of space vectors: If three vectors are not * * * planes, then any vector in space has a real array, so.
38. If three vectors are not * * * planes, then the set of all spatial vectors is
This set can be regarded as generated by vectors,
This is called the bottom of space, and this is called the bottom vector. Any three vectors with non-* * faces in a space can form a base of the space.
39. Let,, are three pairs of vertical unit vectors, which have a common starting point (call them unit orthogonal bases). Taking the common starting point of,, as the origin, and the direction of,, as the axis, the axis and the positive direction of the axis respectively, a spatial rectangular coordinate system is established. Any vector in space can be translated to make its starting point coincide with the origin, and a vector can be obtained.
40, if, then.
.
.
.
If it is a non-zero vector, then.
If so, then.
.
.
, and then.
4 1, based on a certain point in space, then the position of any point in space can be represented by a vector. This vector is called the position vector of the point.
42. The position of any straight line in space can be determined by the last fixed point and a fixed direction. A point is a point on a straight line, and a vector represents the direction vector of the straight line, so that there is any point on the straight line, so that the point and vector can not only determine the position of the straight line, but also specifically represent any point on the straight line.
43. The position of a plane in space can be determined by two intersecting straight lines inside. Let these two intersecting lines intersect at a point, and its direction vector is. For any point on the plane, there is an ordered real number pair, so that the point and vector can determine the position of the plane.
44, the straight line is vertical, take the direction vector of the straight line, and the vector is called the normal vector of the plane.
45. If the space does not coincide with two straight lines, the direction vectors of are,, and respectively.
,.
46. If the direction vector of a straight line is 0 and the normal vector of a plane is 0, then
,.
47. If the normal vectors of two planes that do not overlap in space are,, respectively, then
,.
48. Let the included angle of straight lines in different planes be, the direction vector be, and the included angle be, then there is.
.
49. Let the direction vector of a straight line be, the normal vector of a plane be, the angle formed by, and the included angle between, have.
50, let it be the normal vector of two faces of dihedral angle, then the included angle (or other angles) of the vector is the size of the plane angle of dihedral angle. If the plane angle of the dihedral angle is, then.
5 1, the distance between points can be converted into the modulus calculation of the corresponding vector between two points.
52. Find a point in a straight line. If the vector passing through a fixed point and perpendicular to the straight line is, the distance from the fixed point to the straight line is.
53. A point is a point out of the plane, a point in the plane, and the normal vector of the plane, so the distance from the point to the plane is.