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How to prove that a vector is the normal vector of a plane
It is proved that any vector of a plane multiplies its point by 0
You can set this plane as the X-Y plane
Any vector coordinate can be written as (x, y)
How to find the plane normal vector, please specify the specific method!
Vector BA=(1,0,-1), Vector BC=(0,1,1)
Try vector p=(a, y, z)
P is perpendicular to BA and BC
X-z=0, y+z=0
X=-y=z
Take a set of non-zero solutions, x=1, y=-1, z=1
Normal vector (1,-1,1)
After you know the 2 vectors in the plane, both are vertical,,
Proving Pythagorean Theorem by Using Vector
If the oblique edges of a right triangle are regarded as a vector on a two-dimensional plane and the two oblique edges are regarded as a projection on the coordinate axis of a plane rectangular coordinate system, the meaning of the Pythagorean theorem can be investigated from another angle, that is, the square of the vector length is equal to the sum of the squares of the projection lengths on a set of orthogonal bases in the space where it is located.
If the oblique edge of a right triangle is regarded as a vector on a two-dimensional plane and the two oblique edges are regarded as a projection on the coordinate axis of a plane rectangular coordinate system, the meaning of the Pythagorean theorem can be investigated from another angle, that is, the square of the vector length is equal to the sum of the squares of the projection lengths on a set of orthogonal bases in the space where it is located.
How many methods of proving Pythagorean theorem? How many ways to prove Pythagorean theorem?
There are hundreds of methods to prove Pythagorean theorem. The most classical Chinese method is to draw two squares with sides of (a+b), as shown in the figure, where a and b are right-angled sides and c is oblique side. These two squares are congruent, so their areas are equal. The left and right diagrams each have four triangles congruent with the original right-angled triangle, and four left and right triangles.
There are hundreds of methods to prove Pythagorean theorem. The most classical Chinese method is to draw two squares with sides of (a+b), as shown in the figure, where a and b are right-angled sides and c is oblique side. The two squares are congruent, so their areas are equal. The left and right figures have four triangles congruent with the original right-angled triangle, and four left and right triangles respectively.
Method of Proving Pythagorean Theorem I don't want a right triangle, I want a right triangle
. Chinese method: Draw two squares with (a+b) sides, as shown in the figure, where a and b are right-angle sides and c is oblique side. The two squares are identical, so the area is equal.
Each of the left and right figures has four triangles that are identical to the original right triangle. The sum of the area of the left and right triangles must be equal. Remove the four triangles from the left and right figures, and the area of the rest of the figure must be equal. The left figure has two squares, with a and b as sides respectively. The right figure has a square with c as sides.
A^2+b^2= c^2.
This is the method introduced in our geometry textbook. It is intuitive and simple, and anyone can understand it.
2. Greek method: draw a square directly on the three sides of a right triangle, as shown in the figure.
It is easy to see that,
△ABA' AA' C.
Via the C-direction A "B" lead, cross AB at C ", cross A "B" at C ".
△ABA' is the same height as square ACDA', the former area is half of the latter area,△AA' C is the same height as rectangular AA'' C', the former area is also half of the latter area. From △ABA' AA'' C, we know that the area of square ACDA' is equal to the area of rectangular AA'' C' C.
S square AA'' B'' B = S square ACDA'+ S square BB' EC,
I.e. a2+b2= c2.
As for the triangular area is the same as the bottom of the rectangular area half, can be obtained by the cut-and-fill method (please prove it).
This is the proof of the ancient Greek mathematician Euclid in his Geometric Primordial.
The reason why the above two proof methods are wonderful is that they only use two basic concepts of area with few theorems:
(1) Equivalence of congruent areas;
(2) A graph is divided into several parts, and the sum of areas of each part is equal to the area of the original graph.
This is a perfectly acceptable idea of simplicity that anyone can understand.
Chinese mathematicians of past dynasties have various methods of proving Pythagorean theorem, and they have made many illustrations for Pythagorean theorem, among which Zhao Shuang (i.e. Zhao Junqing) proved it earlier in his paper Pythagorean Diagram Annotation, which is attached to Zhou Gui Suan Jing.
As shown in the figure, the four right-angle triangles in the figure are painted with vermilion, and the small square in the middle is painted with yellow, which is called medium yellow solid, and the square with the chord as the side is called string solid, and then after the complement and collocation," make the difference complement each other, according to its category ", he affirmed that the relationship among the three Pythagorean strings is in accordance with the Pythagorean theorem, i.e.," Pythagorean each multiplies and is the string solid, and the square division is the chord ".
Zhao Shuang's proof of the Pythagorean theorem shows that Chinese mathematicians have excellent proof ideas.
Many western scholars have studied Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean
A normal vector of line 2(x-2)-3(y+1)=0 N=(a, a-2) Find a.-. A normal vector of line 2(x-2)-3(y+1)=0 N=(a, a-2) Ask a.-.
One normal vector =(2,-3)
Another normal vector N=(a, a-2) is obtained from X1Y2=X2Y1
A*(-3)=2(a-2)=4/5
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