Question

Let v be a vector in Rn, then ||v|| is a scalar.
Select one:
a. False
b. True

Answer 1

the answer is (b) true

Let v be a vector in Rⁿ in the form $v=(v_{1}, v_{2}, v_{3}, . . . . ., v_{n})$ ,

then

$|| v_{1} ||^{2}= (v_{1})^{2}+ (v_{2})^{2}+(v_{3})^{2}+ . . . . .+(v_{n})^{2}$

since $v_{1}, v_{2}, v_{3}, . . . . ., v_{n}$ are real numbers , then

$(v_{1})^{2}+ (v_{2})^{2}+(v_{3})^{2}+ . . . . .+(v_{n})^{2}$ is a positive real number ( scalar )

Thus $|| v_1 ||= \sqrt{v_1^2+ v_{2}^2+v_{3}^2+ ...+v_n^2}$ is a positive real number,

i.e. $|| v_{1} ||$ is scalar.