Ask for help with some questions about high school space solid geometry vectors must, detailed addi

Let's look at the figure, the normal vector, divided by the modulus, is it a unit vector? To be exact, a normal vector with a modulo length of 1.

a=（1,1,1），|a|= root number 3, (each coordinate squared and reopened).

Vectors have both magnitude and direction, the modulus of the vector refers to the size of the vector (also called the length of the vector), and the unit vector refers to the vector with a length of 1, therefore, [a unit vector = a vector The modulus of a certain vector];

The modulus of the vector is equal to the sum of the squares of the coordinates and then the square;

It cannot be said that the normal vector divided by modulus is equal to 1, but is equal to the unit vector in the same direction as the normal vector;

Since the direction of the normal vector is the same as the direction of the distance, the normal vector is found first.

Is the normal vector, divided by the modulus, a unit vector? To be exact, a normal vector with a modulo length of 1.

a=（1,1,1），|a|= root number 3, (each coordinate squared and reopened).

These questions of yours need to be illustrated, the text is not easy to express, really, you can ask your math teacher, he will definitely be able to explain it to you.

I really suggest that you ask your math teacher, these questions of yours need to be illustrated and explained, and the text cannot fully express them.

There are both compulsory and elective courses, the first chapter of compulsory 2 is the preliminary of three-dimensional geometry, and the second chapter of the second chapter of the analytic geometry is only about the spatial coordinate system. Elective 2-1 (Science Book) Chapter 3.

Space vector and solid geometry test points.

1) Take the vector as the carrier, use the linear operation of the vector, especially the application of the quantity product, prove the parallel, perpendicular and other problems, examine the problems with various question types, especially the solution questions, use the space vector quantity product to solve the corresponding geometric problems, establish the appropriate spatial Cartesian coordinate system, and use the vector sitting number to answer the royal standard operation to prove that the line line, line surface, surface surface is parallel to the perpendicular, and the solution problem of space angle and distance, mainly to solve the problem, mostly belongs to the mid-range problem.

2) Use the relevant knowledge of vector quantity product to solve geometric problems, and use vector coordinate operation to examine geometric problems such as parallel, perpendicular, angle, and distance.

The magnitude of the vector is called the length or modulus of the vector

1. A vector with a length of 0 is called a zero vector and is denoted as 0.

2. A vector with modulo 1 is called a unit vector.

3. A vector that is equal in length to vector a but in the opposite direction is called the opposite vector of a. Write it as -a.

4. Vectors with equal directions and equal modularities are called equal vectors.

As a new content, space vector has considerable advantages in dealing with spatial problems, and is more flexible than the original method of dealing with spatial problems.

If the problem of line-surface relationship and the problem of finding angles and distances in solid geometry are transformed into vectors, how to take vectors or establish spatial coordinate systems, find the parallel-perpendicular and other relationships demonstrated, and how to express the angles and distances in vectors is the key to the problem

The calculation and proof of solid geometry often involve two major problems: one is the position relationship, which mainly includes the line and line perpendicular, the line and the plane are perpendicular, the line and the line are parallel, and the line and plane are parallel; The second is the measurement problem, which mainly includes the distance from the point to the line, the point to the surface, the angle formed by the line and the line, and the angle formed by the surface. There are more examples of how to use vectors to prove that lines and planes are perpendicular and calculate line angles, and how to use vectors to prove that lines and surfaces are parallel, and there are not many examples of calculating the distance from the point to the plane, the angle of the line and the angle of the surface, which plays a role of throwing bricks and attracting jade.

The following is a simple common sense to solve with the vector method:

1. The sufficient and necessary condition for the space point p to be located in the plane MAB is that there is a unique ordered real number pair x and y, such that.

Or to space a certain point を has.

2. For any point o in space and three points a, b, c, if:

where x y z=1), then the four points p, a, b, and c are coplanar

3. Use vectors to prove a b, that is, to take vectors on a and b respectively.

5. Use vectors to find the angle between two straight lines A and B, that is, take them on A and B respectively, and find:

Problem 6, using vectors to find distances is transformed into vector modulus problems:

7. The key to using the coordinate method to study the relationship between lines and surfaces or to find angles and distances is to establish a correct spatial Cartesian coordinate system and correctly express the coordinates of known points

Find the truth about the angle between the straight line and the side of space:

1. Find the angle between the straight line vector d and the side normal vector n (theta) The angle between the two vectors is obvious, and the answer will be arccos(x)2, however, this angle is not the angle between the straight line and the side (alfa) In fact: alfa and eata are mutually congruent.

So alfa = pi 2-theta (radian system) thus, theta = 90-alfa

Since the original theta = arccos(x), then x=cos(theta)=cos(pi 2-alfa)=sin(alfa).

So the truth is: alfa=arcsin(x) is what is sought, and arcsin is the more reasonable and direct answer, but the above is only a traditional analytical method.

The charm of spatial geometry lies in its flexibility, different ways of thinking have different solutionsarcsin arccos is just different ways of describing solutions, and if the things described are at the same angle, there is no contradiction.

I wish you all the best in the college entrance examination.

Why do I have to fix to arccos?In the end, the important thing is to get the angle, and the arcsin arccos can get the angle, and there is no reason why it is always arccos.