Problem — True or false on vectors

Problem

State whether the following statements are true or false, justifying your answer: (a) two vectors with the same magnitude are always equal; (b) the sum of two vectors can have a magnitude smaller than either of the two; (c) if a quantity has dimensions $[L\,T^{-1}]$ it is certainly a velocity.

Solution

(a) False. A vector is defined by magnitude, direction and sense: two vectors with the same magnitude can differ in direction or sense. For example $\vv{a} = 3\;\uv{x}$ and $\vv{b} = 3\;\uv{y}$ have the same magnitude $3$ but are different. Equality between vectors requires agreement of all three characteristics.

(b) True. The magnitude of the sum depends on the angle between the vectors:

$\norm{\vv{a}+\vv{b}} = \sqrt{a^2 + b^2 + 2ab\cos\theta}$

If the vectors are nearly opposite ($\theta$ close to $180^\circ$, $\cos\theta$ negative) the sum can turn out smaller than both. Example: $\vv{a} = 5\;\uv{x}$ and $\vv{b} = -4\;\uv{x}$ give a resultant of magnitude $1$, smaller than both $5$ and $4$. In the limit, two opposite vectors give a zero sum.

(c) False. The dimensions $[L\,T^{-1}]$ are a necessary but not sufficient condition for a velocity. Many other quantities share those same dimensions without being a velocity: for example the flow rate of a duct per unit cross-section, or the product of a frequency and a wavelength. Dimensional analysis filters out the possible quantities but does not uniquely identify the physical meaning.

Summary. $\ev{\text{(a) F}\quad\text{(b) T}\quad\text{(c) F}}$

Linked atoms

Argomenti: Vectors Concetti: Components of a vector Competenze: Dimensional analysis