For a given vector v in R^n, choose the correct answer below for the given state

Question

For a given vector v in R^n, choose the correct answer below for the given statement v middot v = ||v||^2. The given statement is false. It is not possible for it to be true because v middot v simplifies to a vector, whereas, ||v||^2 simplifies to a scalar. The given statement is true. By the definition of the length of a vector v, ||v|| = squareroot v middot v. The given statement is false. By the definition of the length of a vector v, ||v|| = v middot v. It follows that ||v||^2 = (v middot v)^2. The given statement is true. By the definition of the length of a vector v, ||v|| = v middot v. It follows that ||v||^2 = (v middot v)^2.

Explanation / Answer

SOLUTION:

Rn is an inner product space with the inner product defined by, <u,v>=u.v for two vectors u and v in Rn.

With respect to this inner product the norm(length) of a vector v is the scalar quantity defined by: ||v||2 = <v,v> = v.v

Therefore the correct statements are:

B and C.

A. incorrect since, v.v is scalar (not a vector).

D. incorrect since, ||v||2=v.v ( (v.v)2)

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