About Position Relationship of Straight Lines A formula for judging the position relationship betwe

Set the coordinates of point b (x, y), and the coordinates of point m in ab can be calculated.

Point b conforms to the bisector equation of the inner angle at point b.

The point m conforms to the midline equation of c.

Two equations can be solved for x,y, which is the coordinates of point b.

Then find the symmetry of point A with respect to the bisector of the inner angle of point B.

According to the symmetry point and the point B, the straight line equation BC also comes.

There are intersecting (perpendicular, non-perpendicular, coincidental) and parallel in the same plane, and in different planes there are different planes perpendicular and different planes parallel.

The positional relationship between the two direct lines can be divided into two categories:

In the same plane:

Parallel, intersecting, coincidental.

Sikai in two planes:

Straight lines on different planes. 2. Two straight lines are parallel and perpendicular.

Judgment. The judgment of two straight lines parallel and perpendicular is divided into two categories, one is the point oblique judgment and the other is the general judgment.

<> analysis: Equations for straight lines parallel and perpendicular to the straight line ax+by+c=0 (satisfying that the sum of squares of the coefficients in front of x and y is not equal to zero).

Can be set to: Parallel:

ax+by+d=0 (where c is not equal to d).

Vertical: bx -ay+m=0

3. Ideas for solving problems.

1) Two straight lines in parallel:

The slope of two straight lines.

Equal and on the axes.

intercept on . Unequal, or the slope of both lines does not exist and the intercepts of the two lines on the x-axis are not equal.

2) Two straight lines perpendicular:

The product of the slopes of two straight lines is equal to -1, or the slope of one line is 0 and the slope of the other line does not exist.

Positional relationship between straight lines: parallel, intersecting (including perpendicular and non-perpendicular), coincident. The relationship between the carry-over position of straight lines in different planes is: heterogeneous (including perpendicular and non-perpendicular).

1. Straight lines and linear properties:

The position relationships between straight lines and straight lines in the same plane are: parallel, intersecting (including perpendicular and non-perpendicular), and coincident. The relationship between the position of straight lines and straight lines in different planes is: heterogeneous surfaces (including perpendicular and non-perpendicular).

2. The derivative meaning of straight lines and straight lines.

Assuming that two straight lines are not parallel, then they must intersect. In this way, the two non-parallel lines form a triangle with the third truncated line. One of the isotope angles becomes the outer angle of the triangular argumentative pose.

Because the outer angle of a triangle is equal to the sum of the two inner angles that are not adjacent to it, i.e., one of the isotope angles is equal to the sum of the other isotope angle and the non-adjacent inner angles. Therefore, one colocation angle in his tomb is not the same as another colocation life.

That is, two straight lines are not parallel and the isotopic angles are not equal, and vice versa must be true.

3. Location relationship:

4. Mathematical Relationships:

1. General formula: applicable to all straight lines.

ax+by+c=0 (where 0 when a and b are different).

y-y0=k(x-x0)

When k does not exist, a straight line can be expressed as.

x=x03, oblique truncated type: a straight line with an intercept of b on the y-axis (i.e., passing (0, b)) and a slope of k.

From the point oblique type, the oblique truncated formula y=kx+b

As with the point oblique form, it is also necessary to consider whether the k exists or not.

4. Intercept type: not suitable for straight lines perpendicular to any coordinate axis.

Knowing that the line intersects (a,0) with the x-axis and (0,b) with the y-axis, then the line can be expressed as.

bx+ay-ab=0

In particular, when ab is not 0, the oblique truncation can be written as x a+y b=1

5. Two-point formula: straight line through (x1, y1) (x2, y2).

y-y1) (y1-y2) = (x-x1) (x1-x2) (slope k needs to exist).

6. Normal type.

xcosθ+ysinθ-p=0

where p is the distance from the origin to the straight line, and is the angle between the normal and the positive direction of the x-axis.

7. Point direction (x-x0) u=(y-y0) v

u,v is not equal to 0, i.e., the point direction formula cannot represent the equation parallel to the coordinates).

8. Point normal type.

a(x-x0)+b(y-y0)=0

9. General formula.

ax+bz+c=0,dy+ez+fc=0

10. Point-oriented type:

Let the straight direction vector be (u,v,w) and pass through the points (x0,y0,z0).

x-x0)/u=(y-y0)/v=(x-x0)/w

11. X0Y type.

x=kz+b,y=lz+b

The position relationship between straight lines and straight lines in the same plane is: parallel, intersecting (including perpendicular and non-perpendicular), and coincident. The relationship between the position of straight lines and straight lines in different planes is: different surfaces (including vertical and non-perpendicular).

Assuming that two straight lines are not parallel, then they must intersect. In this way, the two non-parallel lines form a triangle with the third truncated line. One of the isotope angles becomes the outer angle of the triangle.

Because the outer angle of the triangular plum shape is equal to the sum of the two inner angles that are not adjacent to it, i.e., one of the isotope angles is equal to the sum of the other isotopic angle and the non-adjacent inner angles. So, one of the isotope angles is not equal to the other.

That is, two straight lines are not parallel and the isotopic angles are not equal, and vice versa must be true.

Nature of parallel lines:1. Straight lines parallel to the same straight line are parallel to each other;

2. Two parallel straight lines are truncated by the third straight line, and the isotopic angles are equal;

3. The straight line of the two flat and rotten rows is truncated by the third straight line, and the internal wrong angle is equal;

4. The two parallel straight lines are truncated by the third straight line, which complements the inner angles of the side.

Parallel, i.e., the slopes are equal, and perpendicular i.e., the slopes are multiplied = -1

The intersection point is the solution of the binary one-dimensional equation, and the equation of the two straight lines in the ontology is connected to obtain a=(7,-3).

Because it is parallel to l3, k=-4, let the linear equation be; y=-4x+b, bring in the point a, find b=25, so the linear equation is: y=-4x+25

The slope of l4 is -2 3, so the slope of the straight line is 3 2

Let the linear equation be: y=3 2x+b2, and bring in the point a, and obtain b2=-27 2, so the linear equation is: y=3 2x-27 2

This type of question is divided into two parts...

1 Intersect and find the intersection point.

Directly connect the relevant linear equations to solve the binary system of linear equations.

2 Parallel and perpendicular.

The linear equation is uniformly transformed into an oblique truncated formula: y=kx+b, and the two ks are equal and parallel.

The product of k is minus 1, i.e. vertical.

If you solve a first, if you are parallel, a1b2=a2b1, and vertically a1a2+b1b2=0

Relation of straight lines in the plane: 1Parallel, 2Intersect, 3Vertical. Relationships between heterogeneous straight lines:1Parallel, 2Vertical.

Parallel, intersecting, coincidental, heterogeneous.

It is determined by the slope of these two straight lines.

Because the slope of the straight line y=-x+2 is -1, and the slope of the straight line y=x is: 1

So the product of the slopes of these two lines is equal to -1, so the position of these two lines is perpendicular to each other.

Whether in the same plane or in the same space, the position relationship between two straight lines intersects, parallels, and overlaps.

Use the formula to judge:

1) Each straight line can be represented using a binary one-dimensional equation.

2) Combine the two straight lines that require the relationship to solve the system of equations.

3) If the two equations are the same, it means that the two straight lines overlap, if the subtraction is a constant after simplification, it means that the two straight lines are parallel, and if the system of equations can find the values of x, y, it means that the two straight lines have an intersection point, indicating that the two straight lines intersect.

It's not parallel, so it's intersecting.

Looking at the slope again, k1·k2 = -1, so it's vertical.