Vectors and Scalars — Question 1

Question 1

A minus sign does not make a quantity a vector

A laboratory report contains these exact entries for a cart on a straight track: Mass is $2.0$ kg, temperature is $-5.0^\circ$C, distance traveled is $12$ m, displacement is $4.0$ m west, instantaneous speed is $3.0$ m/s, and instantaneous velocity is $3.0$ m/s west. A scalar is specified by a numerical value and unit; it has no spatial direction. A nonzero vector additionally has a direction. A vector’s magnitude is a nonnegative scalar. The track reference is fixed to the laboratory.

1. Classify each listed quantity as scalar or vector, explaining why the negative temperature is not a westward quantity.

2. A student deletes “west” wherever it appears. Which resulting entries still specify the original quantity completely? What do the incomplete entries now specify?

3. Why can units alone not distinguish distance from displacement, or speed from velocity? Give two distinct velocities consistent with a speed of 3.0 m/s on this track.

4. If east is chosen positive, the velocity is written $v=-3.0\ \mathrm{m/s}$. Is its magnitude negative? Explain how one signed number can encode a vector on a known line.

Question 1 - Solution

A minus sign does not make a quantity a vector

(a) Classify the physical information being reported.

$$\vec d=-4\hat i\,\mathrm m,\quad\vec v=-3\hat i\,\mathrm{m/s};\qquad m=2\,\mathrm{kg},\ T=-5^\circ\mathrm C,\ s=12\,\mathrm m$$ Mass, temperature, distance and speed are scalars; displacement and velocity are vectors. Celsius temperature can be negative without referring to a spatial direction. Take east as positive for the vector expressions.

(b) Removing direction preserves magnitudes, not nonzero vectors.

$$|\vec d|=4\,\mathrm m,\quad |\vec v|=3\,\mathrm{m/s},\quad\vec v\ne|\vec v|$$ Deleting “west” leaves all scalar entries complete. For displacement and velocity, it leaves only the magnitudes; their original nonzero vectors are no longer specified.

(c) Equal dimensions do not imply equal kinds of quantity.

$$[s]=[\vec d]=L,\quad[v]=[\vec v]=LT^{-1},\quad\boxed{\vec v_\pm=\pm3\hat i\,\mathrm{m/s}}$$ The physical definition, not the unit alone, distinguishes these quantities. The two displayed velocities have equal speeds and opposite directions.

(d) A signed one-dimensional value relies on a declared axis.

$$v_x=-3\,\mathrm{m/s},\quad\vec v=v_x\hat i,\quad|\vec v|=|v_x|=3\,\mathrm{m/s}$$ On a declared line, the sign supplies the two-way direction choice. A negative component specifies west; the norm remains nonnegative. One signed number would not specify an arbitrary planar direction.