If a=0, then vector b is collinear with a if and only if there is a unique real number λ, so that b=λa emphasizes the effect of a=0 If a =0, then the necessary and sufficient condition for the vector b to be collinear with a is that there is a unique real number λ, so that b =λa emphasizes the action of a =0

If a=0, then vector b is collinear with a if and only if there is a unique real number λ, so that b=λa emphasizes the effect of a=0 If a =0, then the necessary and sufficient condition for the vector b to be collinear with a is that there is a unique real number λ, so that b =λa emphasizes the action of a =0

Let a have at least a modulus not equal to 0. If | a|=0, then the direction of a is indefinite, there is no discussion meaning of collinear and noncollinear. Because any b can be collinear with it (specify that the direction of 0 is not the same as that of b) or not (specify that the direction of 0 is not the same as that of b).

Let a have at least modulus not equal to 0. If | a|=0, then the direction of a is indefinite, there is no discussion meaning of collinear and non-collinear. Because any b can be collinear with it (specify that the direction of 0 is not the same as that of b) or not (specify that the direction of 0 is not the same as that of b).

Let a have at least modulus not equal to 0. If | a|=0, then the direction of a is indefinite, there is no discussion meaning of collinear and noncollinear. Because any b can be collinear with it (specify that the direction of 0 is not the same as b) or not (specify that the direction of 0 is not the same as b).

In general, the necessary and sufficient condition for vector a‖ vector b is that there exists a real number λ which is not completely zero, R such that λa vector b vector=0 vector Seeking proof In general, the necessary and sufficient condition for vector a‖ vector b is that there exists a real number λ which is not completely zero, R such that λa vector +μb vector Seeking proof

Necessary and sufficient condition
First is sufficiency: vector a‖ vector b so vector a and vector b are opposite or the same direction, so there is λa vector +μb vector =0 vector
If u is zero, vector b may be arbitrary, so vector a is zero
Necessary λ a vector +μ b vector =0 vector has a real number λ which is not completely zero,μ∈ R satisfies a vector = Xb vector (X is not equal to 0)

Given that the nonzero vectors a and b are not collinear, if the vector (ma+b)//(a-nb), then what is the condition that the real number m satisfies Given that non-zero vectors a and b are not collinear, if vector (ma+b)//(a-nb), then what is the condition that real number m, n satisfies

Because (ma+b)//(a-nb), ma+b=λ(a-nb),(m-λ) a+(1+λn) b=0, because a, b are not collinear, m=λ
Therefore, mn=-1

Because (ma+b)//(a-nb), ma+b=λ(a-nb),(m-λ) a+(1+λn) b=0, because a, b are not collinear, so m=λ
Therefore mn=-1

Because (ma+b)∥(a-nb), ma+b=λ(a-nb),(m-λ) a+(1+λn) b=0, because a, b are not collinear, m=λ
Therefore mn=-1

Given the vector a=(1,1), b=(1,-1),|c|=√2, the real number m satisfies c=ma+nb, then the maximum value of (m-1)^2+n^2 is Given the vector a=(1,1), b=(1,-1),|c|=√2, the real number m, n satisfies c=ma+nb, then the maximum value of (m-1)^2+n^2 is

Solution c=ma+nb=m (1,1)+n (1,-1)=(m, m)+(n,-n)=(m+n, m-n)2=|c|2=(m+n)2+(m-n)2 can get m2+n2=1 1-m2=n2≥0, i.e.-1≤m≤1-2≤-2m≤2 0≤2-2m≤4 and (m-1)2+n2=m2+n.

Vector a=(1,1), vector b=(1,1), vector c=(√ cosα,√ sinα), R, real number m, n satisfies ma+nb=c, then (m-3)^2+n^2 is maximum? A, b, c are vectors, m, n are real numbers. Sweat. It's 2 times the root. Cos and sin 2 did n' t come out? But I haven't learned the analytic geometry yet. Vector a=(1,1), vector b=(1,1), vector c=(√ cosα,√ sinα), R, real number m, n satisfies ma+nb=c, then (m-3)^2+n^2 is maximum? A, b, c are vectors, m, n are real numbers. Sweat. It's 2 times the root. Cos and sin 2 did n' t come up? But I haven't learned to analyze geometry yet.

Because ma+nb=c, m+n=√ cosα, m-n=√ sinα. The two expressions are added respectively after the square to obtain m2+n2=1/2. It can be regarded that (m, n) is the point on the circle with the origin as the center and √1/2 as the radius. The maximum value of (m-3)^2+n^2 is obtained. This form can be regarded as the square form of the distance, that is...

If the real number m, n are not zero, and m is not equal to n, the vector a is non-zero, then is the m-multiple vector a parallel to the n-multiple vector a? Why?

Parallel
No matter how many times a vector is multiplied, the direction is constant. A vector with the same direction is a parallel vector