The product of the projection of one vector and another vector on this vector, provided that the starting position is the same.
Given two non-zero vectors a and b, then |a||b|cosθ(θ is the angle between a and b) is called the quantitative product or inner product of a and b, and it is denoted as a B.
The product of two vectors equals the sum of the products of their corresponding coordinates.
That is, if a = (x 1, y 1) and b = (x2, y2), a b = x 1 x2+y 1 y2.
The formula of vector product: a * b = | a || b | cos θ, a and b represent vectors, θ represents the included angle between the starting points of vectors a and b * * *. Obviously, the vector product represents a number, not a vector. ? The product of the projection of one vector and another vector on this vector, provided that the starting position is the same.
Vector projection
Given two non-zero vectors a and b, then |a||b|cosθ(θ is the angle between a and b) is called the quantitative product or inner product of a and b, which is denoted as a B, and the product of the two vectors is equal to the sum of the products of their corresponding coordinates.
That is, if a = (x 1, y 1) and b = (x2, y2), a b = x 1 x2+y 1 y2.
nature
Let a and b be nonzero vectors, then
(1) let e be the unit vector, and the included angle between e and a is θ, then E A = A E = | A || E | cos θ.
② a⊥b= a b=0
③ when a and b are in the same direction, A B = | A | | B | when a and b are opposite, A A = | A | = A or | A | = √ A A.
(4) | A B |≤| A ||| B |, if and only if A and b*** are straight lines, that is, a∨b, the equal sign holds.
⑤ cos θ = a b╱| a || b | (θ is the included angle of vector a.b).
⑥ The product of zero vector and arbitrary vector is 0.
calculate
(1) exchange method:? a b=？ bachelor's degree
(2) Law of number multiplication: (? λa)？ b=？ λ(? a b)=？ One (? λb)
(3) Distribution law: (? a+b)？ c=？ A c+? B.C.
Geometric meaning
(1) The projection of one vector to another.
Let θ be the angle between A and B, then |b|cosθ is called the projection of vector B in the direction of vector A, and |a|cosθ is called the projection of vector A in the direction of vector B..
②? Geometric meaning of a b
Quantity products? A b equals? The length of |? A | and? Really? | direction projection? Product of b|cosθ
★ Note: The projection and the product of two vectors are both quantities, not vectors.
3 product? What is the geometric meaning of a b? The length of |? A | and? Really? | direction projection? The product of b|cos θ.
The Method of Finding Vector Modulus
Formula method, using | a | = and (a b) 2 =| a | 2 2a b+| b | 2, the modular operation of vectors is transformed into scalar product operation;
(2) Geometric method, using the geometric meaning of vectors.
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The method of finding the maximum value (range) of vector modulus;
Algebraic method, the obtained module is expressed as a function of a variable, and then solved by the method of finding the maximum value;
(2) Geometric method (number-shape combination method), to find out the geometric meaning of module representation and solve it by combining the graphics of moving point representation.