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A: error.
B: error.[ Equal vectors need to be equal in length and direction]
C: error.[ A collinear vector is a parallel vector, as long as the position is parallel.]
D: correct.

Given that A, B, C, O are four points in the plane, if there is a real number λ such that the vector oc=λ vector oa+(1-λ) vector ob, then prove that A, B, C are collinear. Given that A, B, C, O are four points in the plane, if there is a real number λ such that the vector oc=λ vector oa+(1-λ) vector ob, then we prove that A, B, C are collinear.

Vector oa-vector ob=vector BA;
Vector oc-vector ob=vector BC
Vector oc=λ vector oa+(1-λ) vector ob
=λ Vector oa-λ vector ob+vector ob
=λ(Vector oa-vector ob)+ vector ob
=λ Vector BA,
Vector BC=λ vector BA,
A, B, C are collinear

Oa-vector ob=vector BA;
Vector oc-vector ob=vector BC
Vector oc=λ vector oa+(1-λ) vector ob
=λ Vector oa-λ vector ob+vector ob
=λ(Vector oa-vector ob)+ vector ob
=λ Vector BA,
Vector BC=λ vector BA,
A, B, C are collinear

Given three vectors in the plane A=(3,2) B=(1,2) C=(4,1) Answer the following questions to find the real number m, n satisfying A=mB+nC

A=mB+nC, then (3,2)=m (1,2)+n (4,1)=(m+4n,2m+n), from which m+4n=3,2m+n=2, m=5/7, n=4/7.

Given that the plane vectors a, b, c satisfy: a⊥c, b*c=-2,|c|=2, if there is a real number λ such that vector c=vector a vector b, then the value of λ is

From the original formula: c-a=λb squared, c2+a2-2ac=λ2b2, i.e.:4+a2=λ2b2(1) c-λb=a squared, c2 2b2-2cλb=a2, i.e.:4 2b2+4λ=a2(2), a & sup...

Given that the three vertices A, B, C and one point P in the plane of the triangle ABC satisfy the vector PA+vector PB+vector PC=0, if the real number λ

Given vector PA + vector PB + vector PC =0
Vector AB=vector PB-vector PA ---(1)
Vector AC=vector PC-vector PA ---(2)
(1)+(2)=> Vector AB + vector AC = vector PB + vector PC-2 vector PA
λ Vector AP=vector PB+vector PC-2 vector PA
-λ Vector PA = vector PB + vector PC-2 vector PA
(2-λ) Vector PA = vector PB + vector PC
(2-λ) Vector PA=-vector PA
(3-λ) Vector PA=0
Because vector PA is not a zero vector,3-λ=0,λ=3.

A necessary and sufficient condition for a vector b to be collinear with a nonzero vector a is that there is and there is only one real number λ, so that b=λa. There are two arguments for proving sufficiency:1. There is and only a real number λ, such that b=λ a then vector b is collinear with nonzero vector a. 2 If there is a real number λ, then vector b and nonzero vector a are collinear. Is that true? A necessary and sufficient condition for a vector b to be collinear with a nonzero vector a is that there is and there is only one real number λ, so that b=λa. There are two arguments for proving sufficiency:1. There is and only a real number λ, such that b=λ a then vector b is collinear with nonzero vector a. 2 If there is a real number λ, then vector b is collinear with nonzero vector a. Is that true? A necessary and sufficient condition for a vector b to be collinear with a nonzero vector a is that there is only one real number λ, so that b=λa. There are two statements of sufficiency:1 There is and only a real number λ, such that b=λ a then vector b is collinear with nonzero vector a. 2 If there is a real number λ, then vector b is collinear with nonzero vector a. Is that true?

The two algorithms in your column are the same. As long as this number exists, it must be unique. The so-called "and only" of statement 1 is actually unnecessary, and statement 2 seems to have this redundant one

The two algorithms in your column are the same. As long as this number exists, it must be unique. The so-called "and only" of statement 1 is actually unnecessary, and statement 2 seems to remove the redundant one