Why are forces vectors

Forces as Vectors : Free-body diagrams of an object on a flat surface and an inclined plane. Forces are resolved and added together to determine their magnitudes and the net force.

Privacy Policy. Skip to main content. The Laws of Motion. Search for:. Vector Nature of Forces. Forces in Two Dimensions Forces act in a particular direction and have sizes dependent upon how strong the push or pull is. In a sense this is so obvious that it's hard to answer because almost anything you do with forces makes use of their vector nature. Displacement is a vector. If you move 2 feet left and 2 feet left again, you have moved 4 feet. Two arrows 2 feet long pointing left added tip to tail are equivalent to one arrow 4 feet long pointing left.

If you move 2 feet left and 2 feet right, you have moved back to the start. This is the same a not moving at all. You can't add rocks this way. Force adds like this. Two small forces to the left are equivalent to a big force to the left. Equal forces left and right are equivalent to no force. This is why force is a vector. Edit - The comments raise a point that I glossed over. This point is usually not raised when introducing vectors. Mathematicians define a vector as things that behave like little arrows when added together and multiplied by scalars.

Physicists add another requirement. Vectors must be invariant under coordinate system transformations. A little arrow exists independently of how you look at it. A little arrow does not change when you turn so it is now facing forward. Equivalently, little arrows do not change if you rotate the arrow so that it faces forward.

This is because space is homogeneous and isotropic. There are no special places or directions in space that would change you or an arrow if moved to a new location or orientation.

If you move away from Earth gravity is different. If this matters, you must move Earth too. By contrast, a scalar is a single number that does not change under coordinate system transformations. Number of rocks is a scalar. The coordinates that describe a vector change when the coordinate system is changed. The left component of a vector is not a scalar. There is a 1-D mathematical vector space parallel to the left coordinate of a vector. If you rotate the coordinate system, it may be parallel to what has become the forward component.

A physicist would not say it is a vector space. A minor nitpick: force is not a vector. Like momentum, it is a covector or one-form , and covariant. You can see this in several ways:. The difference between a vector and covector may not make sense if you're just starting to learn about physics, and for now, knowing that forces can be "added tip to tail" like vectors may be enough for practical calculations.

But it is something you should start to pay attention to as your understanding matures: like dimensional analysis, carefully keeping track of what your physical objects are, mathematically, is helpful both for building deeper understanding, and catching errors.

Acceleration may well be part of a larger structure eg: 2 index tensor under a larger group of transformations including rotations, boosts, strains, and translations. My point being, when you say acceleration or force is a 3-vector or something else , you have to specify for which group of transformations. For example, "acceleration transforms like a 3-vector under rotations", and that is why we call it a 3-vector.

The real answer in my opinion isn't some underlying philosophical arguments about what a force is. The real answer is that thinking of force as a vector gives you a model that satisfies the single most important criterium for any model: it agrees with experiment.

It is also nice and simple, which is an added bonus. Thinking of forces as vectors will allow you to come up with predictions of what happens when you do experiments, specifically experiments where you apply several forces at once. For instance, put a crate on ice, and pull on it using ropes with spring scales embedded in them to measure the magnitude of all forces involved.

Measure and write down all the forces and their directions, think of forces as vectors, and calculate the resultatnat force acting on the crate, which ought to give you a prediction of its acceleration. Then measure its actual acceleration. The two should agree, to within some error. People have done experiments like this, both more and less sophisticated, for a long time, and so far we haven't found anything to indicate that thinking of forces as vectors gives the wrong result.

Thus thinking of forces as vectors will most likely give accurate results the next time we need to calculate a prediction as well. So we learn to think of forces as vectors because it works. And then philosophers can argue about why it works, usually by putting it into the context of a bigger picture, which has also withstood the test of experiments. That being said, there are natural ways to come up with the idea of even considering that force is a vector. Specifically, each force has a direction and a magnitude.

As pointed out in other comments, this doesnt necessarily mean that is must be a vector kinetic energy clearly has a direction and a magnitude, but isn't usually thought of as a vector.

But it is enough to ask whether it could possibly be a vector, and to start designing experiments around that hypothesis. I had this question previously too and spent a good 5 hours on it. In the end, the explanation for this is just that the displacement acts like a vector. And acceleration being the double derivative of it also acts like one. Why does displacement acts like a vector?? Well, it follow the rules of trigonometry and displacements in one direction is independent of the displacement perpendicular to it.

Hence, we define vector concepts to encompass this behavior. Why does displacement follow the rules of trigonometry?? Well, this has been more or less found by observing rather than deriving.

The most fundamental basis of everything in maths is also observation and logic after all. To demonstrate that it really is, you would perform experiments: start by attaching three spring scales like the ones fishermen use to weigh fish to each other at the same point, and pull the other ends of the scales horizontally at degree angles with equal non-zero force F. The configuration is in the beautiful ascii graphic below, and you can tell the forces are equal by looking at the readings on each scale.

You'll also notice that the point of attachment in the middle stays stationary, that is, the net force is zero. If F was a scalar, it would be impossible to add or subtract exactly 3 non-zero Fs in whichever order, and get 0 as a result.

Now that you know that force is not a scalar, you would then try to figure out a way to get the three Fs to add up to zero, and you notice that if you pair the direction of each spring to each F, you can get exactly that:. You'd then conduct further experiments, in various set-ups, and find that in each case, treating force as a scalar paired with a direction gives the correct result, at which point you would feel justified in saying: for the purposes of calculation, force has both a magnitude and a direction.

This lesson will answer those questions. Useful tool: Units Conversion. Although you can designate a force simply as a number or scalar quantity, it is more useful to state it as a vector where you include the direction of the force. Instead of saying that the force is 2 newtons, you would say something like the force is 2 newtons toward the ground.

It is often useful to break a force vector into its components. You can add two or more force vectors that are at angles with respect to each other to create a new force vector.

An example is if a force is moving an object in a given direction and wind applies a force on it at an angle, the new motion will be as if a force was applied in that direction.