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Vector subtraction algorithm? For example, if vector m=(2,3), vector n (0,5), then what is vector m minus vector n?
Corresponding coordinate minus
M-n=(2,3)-(0,5)
=(2-0,3-5)
=(2,-2)
Vector algorithm If it is a coordinate form such as (x1, y1) and (x2, y2), is addition, subtraction, multiplication and division performed in the corresponding coordinate? If the multiplication is (x1x2, y1y2)? If you are given a coordinate system and then you find the combined vector of two vectors, do you find the two vectors first and then add or subtract with geometric knowledge instead of directly adding or subtracting? (Such as Pythagorean theorem)?
Vector multiplication is the multiplication and addition of the corresponding coordinates. For example,(a, b) and (c, d) are multiplied to ac + bd. Vector multiplication is a number.
Vector multiplication is the multiplication and addition of the corresponding coordinates. For example:(a, b) and (c, d) are multiplied to ac + bd. Vector multiplication is a number.
The Calculation Rule of Vector Some Calculation Rules of Vector
1. Addition of vectors
Addition of vectors
The addition of vectors satisfies parallelogram rule and triangle rule.
Addition of vector OB+OA=OC.
A+b=(x+x', y+y').
A+0=0+a=a.
The operation law of vector addition:
Commutative law: a+b=b+a;
Binding law:(a+b)+c=a+(b+c).
2. Subtraction of vectors
If a, b are mutually opposite vectors, then the inverse of a=-b, b=-a, a+b=0.0 is 0
Subtraction of vectors
AB-AC=CB.
Subtract the vector "
A=(x, y) b=(x', y') then a-b=(x-x', y-y').
3. Number multiplication vector
The product of the real number λ and the vector a is a vector denoted by λa, and a = a.
When λ>0,λa is in the same direction as a;
Multiplication of vectors
When λ<0,λa is opposite to a;
When λ=0,λ a =0, the direction is arbitrary.
When a =0,λ a =0 for any real number λ.
Note: by definition, if λa=0, then λ=0 or a=0.
The real number λ is called the coefficient of vector a, and the geometric meaning of multiplier vector λa is to extend or compress the directed line segment representing vector a.
When >1, the directed line segment of vector a extends to the original times in the original direction (λ>0) or the opposite direction (λ<0);
When <1时,表示向量a的有向线段在原方向(λ>0) or ×× opposite direction (λ<0) is shortened to the original times.
The multiplication of numbers and vectors satisfies the following operation law
Combining law:(λ a)·b=λ(a·b)=(a b).
The distribution law of vector to number (the first distribution law):() a= a a.
The distribution law of number to vector (the second distribution law):λ(a+b)=λa b.
The elimination law of number multiplication vector:1 If the real number λ=0 and λa=λb, then a=b.2 If a=0 and λa=μa, then λ=μ.
4. Quantity product of vector
Definition: If two non-zero vectors a, b are given as OA=a, OB=b, then the angle AOB is called the angle between vector a and vector b, and is denoted as < a, b > and 0≤< a, b 〉≤π
Definition: The quantity product (inner product, dot product) of two vectors is a quantity, which is recorded as a·b. If a and b are not collinear, then a·b=|a b cos〈a, b〉; if a and b are collinear, then a·b=+- a b.
The coordinate representation of the quantity product of the vector: a·b=x·x'+y·y'.
ON THE OPERATIONAL LAW OF THE QUANTITATIVE PRODUCT OF VECTOR
A·b=b·a (commutative law);
(λA)·b=λ(a·b)(on the associative law of number multiplication);
(A+b)·c=a·c+b·c (distribution law);
Properties of the Quantity Product of a Vector
A·a=|a| squared.
A⊥b <=> a·b=0.
|A·b a b|.(This formula is proved as follows:|a·b|=|a b cos |Because 0 cos 1,|a·b a b|)
The Main Difference Between the Quantity Product of Vector and the Operation of Real Number
1. The quantity product of a vector does not satisfy the associative law, i.e.(a·b)·c=a·(b·c); for example:(a·b)^2=a^2·b^2.
2. The quantity product of vector does not satisfy the elimination law, that is, from a·b=a·c (a=0), we can not deduce b=c.
3.|A·b|=|a b|
4. By |a|=|b|, we can not deduce a=b or a=-b.
5. Vector product of vector
Definition: The vector product (outer product, cross product) of two vectors a and b is a vector, denoted as a×b (here is not a multiplication sign, but a representation, which is different from "·", and can also be denoted as "∧"). If a and b are not collinear, then the module of a×b is: a×b =|a b sin〈a, b〉; the direction of a×b is perpendicular to a and b, and a, b and a×b form the right-hand system in this order. If a and b are collinear, then a×b=0.
Vector product property of vector:
A×b is the parallelogram area with a and b sides.
A×a=0.
A vertical b a×b=|a||b|.
Vector Product Operation Law of Vector
A×b=-b×a;
(λA)×b=λ(a×b)=a×(λb);
A×(b+c)=a×b+a×c.
Note: Vector has no division," vector AB/vector CD "is meaningless.
6. Mixed product of three vectors
Mixed product of vectors
Definition: given three vectors a, b, c in space, the vector product a×b of vectors a, b, and then the number product (a×b)·c of vector c,
The number resulting from the mixed product of vectors is called three-way
Mixed product of the quantities a, b, c, denoted (a, b, c) or (abc), i.e.(abc)=(a, b, c)=(a×b)·c
Mixed product has the following properties:
1. The absolute value of the mixed product of the three non-co-directional quantities a, b, c is equal to the volume V of the parallelepiped with a, b, c as the edge, and the mixed product is positive when a, b, c constitutes the right-hand system; the mixed product is negative when a, b, c constitutes the left-hand system, i.e.(abc)=εV (ε=1 when a, b, c constitutes the right-hand system;ε=-1 when a, b, c constitutes the left-hand system)
2. The corollary of the above property: The sufficient and necessary condition of the coplanar of the trivectors a, b, c is (abc)=0
3.(Abc)=(bca)=(cab)=-(bac)=-(cba)=-(acb)
4.(A×b)·c=a·(b×c)
The operation law of vector satisfaction
Multiplicative distribution law
Vector formula What is the vector OB if the vector OB is rotated 90 degrees counterclockwise about point O and 2OA + OB =(7,9) Vector formula What is the vector OB if the vector OA is rotated 90 degrees counterclockwise about point O and 2OA + OB =(7,9) Vector formula What is the vector OB if the vector OB is rotated 90 degrees counterclockwise about point O and 2OA+OB=(7,9)
Let OA=(x, y), then OB=(-y, x)(known from the graph)
2OA+OB=(2x-y,2y+x)=(7,9)
Solution: x=23/5, y=11/5
OB=(-11/5,23/5)
What formulas are there for calculating high and medium vectors Who can tell me some high school vector formula? The more, the better. Thank you.
Let vector a=(x1, y1), vector b=(x2, y2)
Then vector a runs parallel to vector b with x1y2=x2y1
Vertical x1x2+y1y2=0
Multiply by x1x2+y1y2
Cos=(vector) a*b\|a b|
=(X1x2+y1y2)\Root (x1^2+y1^2)*Root (x2^2+y2^2)
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