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The usage vector proves two parallel exercises An exercise in which the usage vector proves two sides parallel
1, Prove that the two normal vectors are collinear
2. Prove that the normal vector of one plane is perpendicular to the direction vector of two straight lines in the other plane.
In any case, the method of undetermined coefficients should be used to determine the normal vector of each face
Solution of mathematical proofs - Knowledge by means of vectors It is known that O is any point in the regular triangle ABC. Make vertical line from O to each side BC, CA and AB, and the vertical foot is P, Q and R respectively. Verify that AR+BP+CQ is the fixed value. Solution of mathematical proofs - Knowledge by means of vectors It is known that O is an arbitrary point in the regular triangle ABC, make vertical line from O to each side BC, CA, AB, and the vertical foot is P, Q, R respectively.
First of all, we use the expression vector AB,"·" to express the point multiplication. Because ABC is a regular triangle, we assume that the length of all three sides is 1. According to the definition of vector point multiplication, AR=·. This point multiplication represents the length of AO line segment projection in AB direction, i.e. AR.
There are five vectors, the length of any two vectors is equal to the length of the other three vectors. Verification: The sum of the five vectors is 0 Argument should be comprehensive! There are five vectors, and the length of any two vectors is equal to the length of the other three. Verification: The sum of the five vectors is 0 The demonstration should be comprehensive!
The following letters represent vectors
Abcde
|A+b|=|c+d+e|Square on both sides ==> a^2+b^2+2ab=c^2+d^2+e^2+2cd+2de+2ce
Write the other 4 expressions in turn
Bc dea
Cdeab
De abc
Ea bcd
Add the five expressions together, cancel each other, and decompose the factors.
==>(A+b+c+d+e)^2=0
So |a+b+c+d+e|=0
Can spatial vectors a, b, c not be coplanar? And certify that.
I can't.
Can be three prisms of three prisms! Although trivectors are not coplanar, they are parallel
Consider its inverse proposition:
If a, b, c are parallel, then they must be coplanar
This proposition is false
So is the original proposition.
I. e. can not
I can't.
Can be three prisms of three prisms! Although trivectors are not coplanar, they are parallel
Consider its inverse proposition:
If a, b, c are parallel, they must be coplanar
This proposition is false
So is the original proposition.
I. e. can not
No. No.
Can be three prisms of three prisms! Although trivectors are not coplanar, they are parallel
Consider its inverse proposition:
If a, b, c are parallel, then they must be coplanar
This proposition is false
So is the original proposition.
I. e. can not
Determine the shape of quadrilateral ABCD according to the following conditions, and prove that 1 Vector AD = vector BC 2 Vector AD = vector 1/3 BC 3 Vector AB = vector DC and module of vector AB = module of vector AC
1. A parallelogram because the vector AD = vector BC.
2. It can only be determined as trapezoid. Since vector AD =1/3 vector BC, AD//BC.
3. Diamond, which is proved to be a parallelogram and has AB=AC, so it is a diamond
1. A parallelogram because the vector AD = vector BC.
2. It can only be determined as trapezoid. Since vector AD =1/3 vector BC, AD//BC.
3. Diamond, it is proved to be parallelogram, and has AB=AC, so it is diamond
1. A parallelogram because the vector AD = vector BC.
2. It can only be determined as trapezoid. Since vector AD =1/3 vector BC, AD//BC.
3. Diamond, which is proved to be a parallelogram and has AB=AC, so it is a diamond.
One mathematics, vector collinear problem proof. Let o_a=e_1, ob=e_2, oc=e_3, if there are real numbers λ_1,λ_2,λ_3 such that λ_1e_1_2e_2_3e_3=0 and λ_1_2_3=0, we prove that ABC is collinear. Write down the details of the process, online and so on, thank you. One mathematics, vector collinear problem proof. Let o_a=e_1, ob=e_2, oc=e_3, if there are real numbers λ_1,λ_2,λ_3 such that λ_1e_1_2e_2_3e_3=0, and λ_1_2_3=0, we prove that ABC collinear. Write down the details of the process, online and so on, thank you.
λ1E1+λ2e2+λ3e3=0
I.e.λ1OA 2OB+(-λ1-λ2) OC=0
So λ1(OA-OC)2(OB-OC)=0
I.e.λ1CA=-λ2CB
So CA and CB are collinear, i.e. A, B and C are collinear
RELATED INFORMATIONS
- 1. On the Proof of Vector Coplanarity I didn't seem to remember all the notes the teacher wrote: Vector AC=λ1 vector AB 2 vector AD has no specific requirements for λ1λ2 Which points are coplanar? What is the relationship between point A and BCD?
- 2. If two vectors are added =0, then is the sum of their abscissa equal to 0?
- 3. If A, B are the two internal angles of the acute triangle ABC... a mathematical solution ~~ 1. If A and B are two internal angles of acute triangle ABC, in which quadrant is the point P (cosB-sinA, sinB-cosA)? Why? 2. Given that the coordinate of the point P on the final edge of the acute angle α is (2sin2,-2cos2), then α is equal to? 3. Given f (sinx)= sin (4n+1) x, find f (cosx)=? Three questions for answers
- 4. The known triangle ABC is an acute triangle, and the three internal angles are AB C The known vector p=(2-2sinA,cosA + sin A) q=(1+ sin A. cos A-sin A) If p is perpendicular q, find the magnitude of the internal angle A
- 5. Let X be the normal vector of X plane through three points A (1,2,3) B (2,0,-1) C (3,-2,0) Given X be the normal vector of the X plane through three points A (1,2,3) B (2,0,-1) C (3,-2,0) Given the plane X through three points A (1,2,3) B (2,0,-1) C (3,-2,0) to find the normal vector of X-plane
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- 8. Matrix proof: If the matrix of order n satisfies AA^T=E, it is proved that x^TAx=0. If the matrix of order n satisfies A^T=-A, it is proved that x^TAx=0.
- 9. Let A be a real matrix of m*n and A^TA be a positive definite matrix. It is proved by data that r (A^TA)≤r (n)≤n, but I don't know where this formula comes from, and the formula is r (AB)≤r (A), r (AB)≤r (B),. I've found the problem in the book, and now there's another question: Shouldn't r (A)≤min {m, n}? Why is the formula directly r (n)≤n? Somebody help me.
- 10. Let A be the s×m matrix, B be the m×n matrix, and X be the vector of n-dimensional unknown columns. Let A be the s×m matrix, B be the m×n matrix, and X be the vector of n-dimensional unknown columns. AB has the same rank as B, i.e. r (AB)=r (B) Let A be the s×m matrix, B be the m×n matrix, and X be the vector of n-dimensional unknown columns. Let A be the s×m matrix, B be the m×n matrix, and X be the vector of n-dimensional unknown columns. AB has the same rank as B, i.e. r (AB)= r (B)
- 11. How to prove that a vector is the normal vector of a plane
- 12. What is the projection of a on b when the vector ab is parallel and reversed What is the projection of a on b when the vector ab is parallel and opposite
- 13. 1. If a and b are collinear, b and c are collinear? 2. If the module of vector a+b is equal to the module of vector a-b, is a multiplied by b=0? 1. If a and b are collinear, then a and c are collinear? 2. If the module of vector a+b is equal to the module of vector a-b, is a multiplied by b=0?
- 14. △The lengths of the opposite sides of the three internal angles A, B and C of ABC are a, b and c respectively. Let the vector P=(a+c, b), Q=(b−a, c−a), if P∥ Q, the size of angle C is ______.
- 15. In triangle ABC, a, b and c are opposite sides of A, B and C respectively, and a^3+b^3= c^3, ask whether the triangle is an acute triangle or an obtuse triangle In triangle ABC, a, b, c are opposite sides of A, B, and C, and a^3+b^3= c^3, ask whether the triangle is an acute triangle or an obtuse triangle
- 16. It is known that a, b, c are opposite sides of three inner angles A, B, C of triangle ABC, vector m=(root 3,-1), n=(cosA, sinA). Given that a, b, c are opposite sides of three inner angles A, B, C of triangle ABC, vector m=(root 3,-1), n=(cosA, sinA).
- 17. Why is the absolute value of vector ab less than or equal to the absolute value of vector a multiplied by the absolute value of vector b
- 18. Given a=(2,-1,3), b=(-1,4,-2), c=(7,5, in), if a, b, c are coplanar, then why is the real number in 65/7? Given a=(2,-1,3), b=(-1,4,-2), c=(7,5), if a, b, c are coplanar, then the real number is 65/7.
- 19. Vector b=(-3,1), c=(2,1), if vector a is collinear with c, find the minimum value of modulus of b+a Vector b=(-3,1), c=(2,1), if vector a is collinear with c, find the minimum value of module of b+a
- 20. Given that a, b are two vectors in the same plane, where a =(1,2),|b|= root 5/2 and a+2b is perpendicular to 2a-b, find the angle between a and b?