Question

Which of the following vectors are unit vectors with respect to the inner product: <(x1, x2, x3) , (y1, y2, y3)> = 2x1y1 = 2x3y3 in R3?

A. (1, 0, 0)   B. (1, 0, 0)/sqrt(2)   C. (1, 0, 1)/sqrt(2)    D. (1, 1, 0)/2

Select from the following:

1. Only A

2. Only B and D

3. Only A and C

4. All of A, B, C and D

5. None of the above

Thank you!

Answer 1

Unit vector: A vector with a magnitude of one is called a unit vector;

A : (1,0,0) ; Its magnitude = sqrt ( 1² +0² + 0² ) = sqrt (1) = 1;
Thus, A is a unit vector

B: (1/sqrt (2) , 0 , 0 ) ; Its magnitude = sqrt ( (1/sqrt(2) ) ² + 0² + 0² ) = sqrt ( 1/2) = 1 / sqrt(2)
Thus, B is not a unit vector

C: (1/ sqrt(2) , 0 , 1/sqrt(2) ) ; Its magnitude = sqrt ( (1/sqrt(2))² , 0 ² , (1/sqrt(2))² ) = sqrt ( 1/2+1/2) = sqrt (1) = 1
Thus, C is a unit vector

D: (1/2, 1/2, 0) ; Its magnitude = sqrt ( (1/2)² + (1/2)² + 0² ) = sqrt ( 1/4 + 1/4+0) = sqrt ( 1/2) = 1/ sqrt(2)
Thus, D is not a unit vector;

Thus, only A and C are unit vectors;

Thus, option (3) is correct;