# Should you always indicate velocity is a vector?

1. Aug 30, 2011

### autodidude

If the question doesn't specify any directions, and the final answer is just a magnitude...(or should you indicate 'in the positive direction' anyway?), do you still put a line over the v?

I also notice in books, the initial and final velocities don't have the line/arrows over the top to show it's a vector, it's not bolded either.

If you use change in speed over change in time to find acceleration (question uses speed but is implied that it's heading in one direction only), is acceleration then a scalar?

2. Aug 30, 2011

### tiny-tim

hi autodidude!

i think the answer is that vectors (by definition! ) exist in a vector space,

and a vector space is defined as including elements called scalars

if S is a vector subspace of a vector space V, and s is in S, and λ is a scalar, then λs is also in S

in a real vector space, for example, the scalars are all the real numbers (including negative numbers)

so in a one-dimensional problem (eg a projectile going straight up and down), everything is in a one-dimensional subspace, and we are perfectly entitled to describe everything by scalars, and some of those scalars can be negative

and of course if we describe them as scalars, then we write them without a line on top (or arrow, or other vector notation)
if distance is a scalar, then speed and acceleration are also scalars

3. Aug 30, 2011

### robphy

While I tend to agree, it is somewhat common practice in physics textbooks
to write the [non-negative] " magnitude of $\vec v$ " or " magnitude of $\bf\mbox{v}$ " (a.k.a. "the speed") as $v$,
rather than the unambiguous but notationally-cumbersome $\left\|\vec v\right\|$.

So, one may wish to write $v_x$ ("x-component of the velocity-vector") (which is a signed quantity).

The bottom line: try to be unambiguous without being too notationally-cumbersome... using phrases or complete sentences, if needed.

(Personally, I like to indicate the vector nature of an equation if it is also true beyond the 1-D case.)

my \$0.02

4. Sep 1, 2011

### autodidude

What does the || || around the v mean? And "x-component of the velocity-vector"? Also, signed quantity? The book doesn't talk about the notation (or I haven't come across it yet, I will check)

5. Sep 9, 2011

### robphy

The double bars mean "magnitude" or "norm" of the vector.
Some texts use single bars (like absolute-value).
http://en.wikipedia.org/wiki/Norm_(mathematics)

If the x-axis is horizontal,
then "x-component of the velocity-vector" means "horizontal speed".
http://www.physicsclassroom.com/class/vectors/u3l2d.cfm

A "signed quantity" means that this quantity could take negative values,
as opposed to (e.g.) speed [the "magnitude of the velocity"],
which is non-negative (i.e. positive or zero).