20.3 nature of vectors (ESAGN)

Vectors are mathematical objects and we will currently study some of their math properties.

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If 2 vectors have actually the very same magnitude (size) and the same direction, climate we call them equal to every other. Because that example, if we have two forces, \(\vecF_1 = \text20\text N\) in the increase direction and also \(\vecF_2 = \text20\text N\) in the increase direction, then we can say that \(\vecF_1 = \vecF_2\).

Equality that vectors

Two vectors space equal if they have the same magnitude and also the same direction.

Just like scalars which can have optimistic or an unfavorable values, vectors can also be confident or negative. A an adverse vector is a vector i m sorry points in the direction opposite to the reference positive direction. Because that example, if in a particular situation, we specify the increase direction together the reference confident direction, then a force \(\vecF_1 = \text30\text N\) downwards would certainly be a negative vector and also could additionally be composed as \(\vecF_1 = -\text30\text N\). In this case, the negative sign (\(-\)) suggests that the direction of \(\vecF_1\) is the opposite to the of the reference positive direction.

an adverse vector

A an adverse vector is a vector that has actually the opposite direction to the reference hopeful direction.

Like scalars, vectors can likewise be added and subtracted. We will investigate exactly how to carry out this next.

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Addition and subtraction the vectors (ESAGO)

including vectors

When vectors space added, we need to take right into account both your magnitudes and directions.

For example, imagine the following. You and also a friend space trying to relocate a hefty box. You stand behind it and push forwards v a pressure \(\vecF_1\) and your friend stands in front and pulls it in the direction of them v a force \(\vecF_2\). The two pressures are in the same direction (i.e. Forwards) and so the complete force exhilaration on the box is:


It is an extremely easy to understand the principle of vector addition through an task using the displacement vector.

Displacement is the vector which describes the change in one object"s position. That is a vector the points indigenous the initial position to the last position.

Adding vectors


masking tape


Tape a line of masking ice horizontally throughout the floor. This will certainly be your beginning point.

Task 1:

Take \(\text2\) steps in the front direction. Usage a item of masking ice cream to mark your end point and label it A. Climate take another \(\text3\) steps in the forward direction. Use masking ice cream to mark your final position together B. Make certain you shot to keep your steps all the exact same length!

Task 2:

Go earlier to your beginning line. Now take \(\text3\) measures forward. Usage a item of masking ice cream to note your end point and label it B. Climate take one more \(\text2\) procedures forward and also use a brand-new piece of masking ice to note your last position as A.


What carry out you notice?

In Task 1, the first \(\text2\) steps forward stand for a displacement vector and the second \(\text3\) measures forward also form a displacement vector. If us did not avoid after the first \(\text2\) steps, we would have actually taken \(\text5\) procedures in the front direction in total. Therefore, if we add the displacement vectors for \(\text2\) steps and \(\text3\) steps, we should obtain a total of \(\text5\) steps in the forward direction.

It does not matter whether you take it \(\text3\) measures forward and then \(\text2\) measures forward, or 2 steps complied with by one more \(\text3\) measures forward. Your last position is the same! The stimulate of the enhancement does no matter!

We can represent vector addition graphically, based on the activity above. Attract the vector for the very first two steps forward, followed by the vector through the following three steps forward.


We add the 2nd vector at the end of the first vector, since this is whereby we currently are after the very first vector has actually acted. The vector indigenous the tail of the very first vector (the beginning point) to the head that the 2nd vector (the end point) is then the sum of the vectors.

As you can convince yourself, the bespeak in i beg your pardon you include vectors does not matter. In the instance above, if you decided to an initial go \(\text3\) steps forward and then an additional \(\text2\) actions forward, the end an outcome would still be \(\text5\) steps forward.

subtracting vectors

Let"s go back to the difficulty of the heavy box that you and your friend room trying come move. If friend didn"t interact properly first, you both could think that you need to pull in your own directions! Imagine you was standing behind the box and also pull it in the direction of you through a pressure \(\vecF_1\) and also your friend stands in ~ the former of the box and also pulls it in the direction of them v a force \(\vecF_2\). In this situation the two forces are in opposite directions. If we specify the direction her friend is pulling in as positive then the force you room exerting should be negative since it is in the contrary direction. We can write the complete force exerted on package as the sum of the separation, personal, instance forces:


What you have done right here is actually to subtract 2 vectors! This is the very same as including two vectors which have actually opposite directions.

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As us did before, we deserve to illustrate vector individually nicely making use of displacement vectors. If you take it \(\text5\) actions forward and also then subtract \(\text3\) measures forward you are left with only two procedures forward:


What did girlfriend physically do to subtract \(\text3\) steps? You originally took \(\text5\) procedures forward yet then you took \(\text3\) steps backward come land up ago with only \(\text2\) measures forward. The backward displacement is stood for by an arrowhead pointing to the left (backwards) with size \(\text3\). The net an outcome of adding these two vectors is \(\text2\) steps forward:


Thus, subtracting a vector from an additional is the same as adding a vector in the opposite direction (i.e. Subtracting \(\text3\) procedures forwards is the very same as adding \(\text3\) actions backwards).