Vector Notation: Struggling to Reconcile Directions

[quincyboy7]

alright let's say we have a vector expressed as another vector times a constant i.e.

F=ma.

But what if we want to signify that a is in the negative direction? Our two possibilities are F=-ma and F=-ma.

The first makes no sense because a vector can't be the product of 2 scalars and the second makes no sense because it contradicts the general rule shown above. I'm really struggling to reconcile vector notation with the concept of picking a direction...what gives?

 

[praharmitra]

Well, there are two situations you are talking about.

1. There are two different vectors F and a (i'll not write them in bold in future), with F = ma. If you want to signify that F and a point in the same direction, then the scalar factor m > 0. However, if you want to signify that they point in the opposite direction, we simply define the scalar factor m < 0.

NOTE: The equation you have written, (F= ma) is the famous Newton's second law, where m > 0 always, signifying the fact that the acceleration of a particle is ALWAYS in the direction of net force.

2. Picking a direction:

To understand vector notation clearly, what you should do is, before beginning any problem of vectors, start by making a coordinate system. Draw two axes, name them x and y. Now draw your vectors and use your coordinate axes to give them appropriate values. This way, you won't have to worry about directions or positivity or negativity of vectors, etc.

 

[quincyboy7]

Well, there are two situations you are talking about.

1. There are two different vectors F and a (i'll not write them in bold in future), with F = ma. If you want to signify that F and a point in the same direction, then the scalar factor m > 0. However, if you want to signify that they point in the opposite direction, we simply define the scalar factor m < 0.

NOTE: The equation you have written, (F= ma) is the famous Newton's second law, where m > 0 always, signifying the fact that the acceleration of a particle is ALWAYS in the direction of net force.

2. Picking a direction:

To understand vector notation clearly, what you should do is, before beginning any problem of vectors, start by making a coordinate system. Draw two axes, name them x and y. Now draw your vectors and use your coordinate axes to give them appropriate values. This way, you won't have to worry about directions or positivity or negativity of vectors, etc.

Taking F=ma, I understand that m>0, but what if the acceleration vector is in the negative direction (like gravity's acceleration?). Should I write F=m(-a) or F=m(-a)?

 

[gneill]

You don't need to do anything special. The vector a embodies the direction in its components. Thus F = ma confers the appropriate direction to the force vector F without any further tinkering with signs.

 

[cryora]

For example, you have F=ma where a is pointing down towards the ground.

You would not need to add a sign until you establish a coordinate system where, for instance, y represents a vertical axis, and positive y means up while negative y means down, as well as a scalar quantity that represents direction in this axis only.

Then you will have a different symbol representing the scalar y component of the vector F - let's call this scalar Fy. If F is pointing directly in the negative y direction, then its scalar y component, Fy, will equal m(-a). A scalar has no direction, but it can take on any value, including negatives, but when it is used to represent a linear component of a vector, then the negative symbol can imply a "negative" direction.