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If I have A....p....B....d....C points

  1. If I am initially on B and walk towards c, it's a positive displacement. Example: BC=10m east and then all of sudden I change my vector and walk to d. Is it a negative displacement?

Please help in direction of displacement, its confusing. I usually get confused when the displacement is + or - and which directions we should use.

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Displacement is a vector. A vector has a direction, not a sign.

It is frequently convenient to choose a coordinate system where vectors to the east are represented with positive numbers and vectors to the west with negative numbers (or vice-versa). That lets you use simple arithmetic to decide that, if you go 10m to the east and then 15m to the west, you are 5m to the west of your starting position. If you have previously decided that “to the west” is negative, then “5m to the west” is negative.

In a comment elsewhere, you suggest the phrasing “negative 5m to the west.” This double negative is begging for confusion. If you are really set on including a negative sign, you would say “negative 5m in a coordinate system where ‘positive’ means ‘east.’” There ain’t nobody who was never not confused by no double negatives.

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  • $\begingroup$ So it won't be appropriate to say -5m west, its either 5mwest or -5 m? $\endgroup$
    – Aarya Chavan
    Commented Jun 24, 2022 at 14:46
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    $\begingroup$ Correct. Another way to think of it is that a vector has a magnitude and a direction, defined so that the magnitude is always positive. The “negative west direction” is the east. $\endgroup$
    – rob ♦
    Commented Jun 24, 2022 at 16:07
  • $\begingroup$ Thank you so much $\endgroup$
    – Aarya Chavan
    Commented Jun 24, 2022 at 21:09
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If you are considering the displacement from C to d, then yes, this is a negative displacement. However, if you are considering the total displacement (i.e. the one defined with A being the initial point and d being the final one), then no, the displacement is still positive, as d lies on the right hand side of A (in the way you depicted things). If anything is still unclear, please do not hesitate to comment.

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  • $\begingroup$ so if distance between c and d is 5m, and i traveled from c to d i can say my displacement is -5m west? $\endgroup$
    – Aarya Chavan
    Commented Jun 24, 2022 at 8:08
  • $\begingroup$ yes. You can also evaluate it algebraically, by assigning a position to points C and d, like $x_c=1005\ m$ and $x_d=1000\ m$. Then the displacement is defined as the final poisition minus the initial one. That is, $\Delta x=x_c-x_d=-5\ m$. The numbers 1005 and 1000 are merely examples and most often they are given by the respective exercise. $\endgroup$
    – schris38
    Commented Jun 24, 2022 at 8:40
  • $\begingroup$ I simply used them to demonstrate the effectiveness of the definition algebraically! So, in this way the sign is manifest. $\endgroup$
    – schris38
    Commented Jun 24, 2022 at 8:41
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Start by defining a unit vector (positive direction) eg $\hat e $(ast) which in this case might be to the right in the diagram below and $\hat w$(est) to the left.
This is just equivalent to using the words east and west to define the directions.

enter image description here

$p$ and $d$ are distances between positions $A$ and $B$ and $B$ and $C$ respectively.

Assume that $\vec {AB}$ is shorthand for the displacement from position $A$ to position $B$ and that that $\vec {BA}$ is shorthand for the displacement from position $B$ to position $A$.

$\vec {AB}$ = $-\vec {BA}$, so moving from position $A$ to position $B$ and then back to position $A$ the total displacement is $\vec {AB}+\vec {BA} = \vec {AB} + (-\vec {AB}) = \vec 0$

$p$ is the magnitude of the displacement $\vec {AB}$ and also the displacement $\vec {BA}$.

$\hat e = - \hat w$

We can write $\vec {AB}= p \,\hat e$ where $p$ is the component of displacement in the $\hat e$ direction but also $\vec {AB} = p\, (-\hat w) = -p\, \hat w$.
Thus $-p$ is the component of $\vec {AB}$ in the $\hat w$ direction.

Suppose we need to find the displacement when moving from position $A$ to position $C$ and back to position $B$.

$\vec {AC} + \vec {CB}= \vec {AB} = p\, \hat e = -p\,\hat w$.

A longer method is as follows.

$\vec {AC}+\vec {CB} = (p+d)\,\hat e + d\,\hat w = p\,\hat e + d\hat e +(-d\,\hat e) = p\,\hat e$

One could also use west as the positive direction.

$\vec {AC}+\vec {CB} = (p+d)\,\hat e + d\,\hat w = (p+d)\,(-\hat w) +d\,\hat w = -p\,\hat w$

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You define a positive x axis, positive y axis, positive z axis.

E.g -1,0,1

A displacement to the left in this case, is denoted by a negative, and displacement to the right is denoted by a positive.

By definition I can define what west "is"

West $= -1\hat i$

Thus a displacement of $-5\hat i$

$= 5 [-1\hat i]$ $= 5 $ west

Questions like these, will always be a positive displacement, but the vectors direction changes. BC will be a positive displacement, in the direction of c from b

Also -5 west = 5 east

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