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Chapter 1: Problem 51

Mark the correct statement (A) \(|\vec{a}+\vec{b}| \geq|\vec{a}|+|\vec{b}|\) (B) \(|\vec{a}+\vec{b}| \leq|\vec{a}|+|\vec{b}|\) (C) \(|\vec{a}-\vec{b}| \geq|\vec{a}|+|\vec{b}|\) (D) All of the above

Short Answer

Expert verified
The correct statement is: (B) \( |\vec{a}+\vec{b}| \leq|\vec{a}|+|\vec{b}| \)

Step by step solution

01

Statement A

\( |\vec{a}+\vec{b}| \geq|\vec{a}|+|\vec{b}| \) This is the reverse triangle inequality theorem, which states that the magnitude of the sum of two vectors is greater than or equal to the difference of their magnitudes. However, this is not the correct formulation of the theorem. Therefore, statement A is not correct.
02

Statement B

\( |\vec{a}+\vec{b}| \leq|\vec{a}|+|\vec{b}| \) This statement is the triangle inequality theorem, which states that the magnitude of the sum of two vectors is less than or equal to the sum of their magnitudes. To see this, consider that when we add two vectors, we can form a triangle with their tails at a common point and their heads connected: the resulting vector forms the third side of the triangle. Since the sum of the lengths of any two sides of a triangle is always greater than or equal to the length of the third side, statement B is correct.
03

Statement C

\( |\vec{a}-\vec{b}| \geq|\vec{a}|+|\vec{b}| \) This statement states that the magnitude of the vector difference is greater than or equal to the sum of the magnitudes of the individual vectors. This contradicts the triangle inequality theorem, as it implies that any two sides of a triangle can be shorter than the third. Therefore, statement C is not correct.
04

Statement D

All of the above (A, B, and C) Since A and C are not correct, statement D cannot be correct. The correct statement is: (B) \( |\vec{a}+\vec{b}| \leq|\vec{a}|+|\vec{b}| \)

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Most popular questions from this chapter

Two forces \(\vec{F}_{2}=500 \mathrm{~N}\) due east and \(\vec{F}_{2}=250 \mathrm{~N}\) due north have their common initial point. \(\vec{F}_{2}-\vec{F}_{1}\) is (A) \(250 \sqrt{5} \mathrm{~N}, \tan ^{-1}(2) \mathrm{W}\) of \(\mathrm{N}\) (B) \(250 \mathrm{~N}, \tan ^{-1}(2) \mathrm{W}\) of \(\mathrm{N}\) (C) Zero (D) \(750 \mathrm{~N}, \tan ^{-1}(3 / 4) \mathrm{N}\) of \(\mathrm{W}\)

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