20.3 properties of vectors (ESAGN)

Vectors are mathematical objects and also we will now study few of their math properties.

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

Equality of vectors

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

Just like scalars which deserve to have positive or an adverse values, vectors can also be hopeful or negative. A negative vector is a vector i beg your pardon points in the direction opposite to the reference positive direction. For example, if in a certain situation, we define the upward direction as the reference positive direction, then a force \(\vecF_1 = \text30\text N\) downwards would be a negative vector and also could additionally be created as \(\vecF_1 = -\text30\text N\). In this case, the an unfavorable sign (\(-\)) shows that the direction of \(\vecF_1\) is the opposite to that of the reference confident direction.

negative vector

A an unfavorable vector is a vector that has actually the opposite direction to the reference optimistic direction.

Like scalars, vectors can additionally be added and subtracted. We will certainly investigate how to do this next.

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Addition and subtraction that vectors (ESAGO)

including vectors

When vectors are added, we should take right into account both your magnitudes and directions.

For example, imagine the following. You and also a friend are trying to move a heavy box. You was standing behind it and also push forwards v a force \(\vecF_1\) and your girlfriend stands in front and pulls it in the direction of them through a pressure \(\vecF_2\). The two forces are in the same direction (i.e. Forwards) and also so the total force acting on the box is:


It is an extremely easy to know the concept of vector enhancement through an activity using the displacement vector.

Displacement is the vector which explains the readjust in one object"s position. The is a vector the points from the initial position to the final position.

Adding vectors


masking tape


Tape a line of masking tape horizontally throughout the floor. This will be your starting point.

Task 1:

Take \(\text2\) procedures in the front direction. Usage a piece of masking ice cream to note your end point and label it A. Climate take an additional \(\text3\) measures in the front direction. Use masking ice to note your final position together B. Make sure you try to store your measures all the same length!

Task 2:

Go ago to your beginning line. Now take \(\text3\) actions forward. Usage a item of masking ice cream to note your end suggest and label it B. Climate take an additional \(\text2\) actions forward and use a brand-new piece of masking tape to mark your last position as A.


What do you notice?

In Task 1, the first \(\text2\) procedures forward represent a displacement vector and also the second \(\text3\) procedures forward also type a displacement vector. If we did not stop after the an initial \(\text2\) steps, we would have taken \(\text5\) measures in the forward direction in total. Therefore, if we include the displacement vectors because that \(\text2\) steps and also \(\text3\) steps, we should acquire a complete of \(\text5\) procedures in the front direction.

It walk not issue whether you take it \(\text3\) procedures forward and then \(\text2\) actions forward, or 2 steps followed by an additional \(\text3\) steps forward. Your final position is the same! The order of the enhancement does not matter!

We deserve to represent vector addition graphically, based upon the activity above. Attract the vector for the very first two procedures forward, complied with by the vector through the next three procedures forward.


We include the 2nd vector in ~ the finish of the an initial vector, due to the fact that this is wherein we currently are after the first vector has acted. The vector indigenous the tail the the very first vector (the beginning point) come the head the the 2nd vector (the end point) is climate the amount of the vectors.

As you have the right to convince yourself, the bespeak in which you include vectors does not matter. In the example above, if you determined to an initial go \(\text3\) procedures forward and then another \(\text2\) steps forward, the end an outcome would still be \(\text5\) actions forward.

subtracting vectors

Let"s go back to the problem of the heavy box that you and your friend space trying come move. If friend didn"t communicate properly first, girlfriend both can think the you must pull in your own directions! Imagine you stand behind the box and pull it in the direction of you v a force \(\vecF_1\) and your friend stands in ~ the former of the box and also pulls it in the direction of them with a force \(\vecF_2\). In this instance the two pressures are in opposite directions. If we define the direction your friend is pulling in as positive then the force you room exerting should be negative since it is in the contrary direction. We deserve to write the complete force exerted on package as the amount of the separation, personal, instance forces:


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

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As we did before, we can illustrate vector individually nicely making use of displacement vectors. If you take it \(\text5\) actions forward and then subtract \(\text3\) measures forward you space left with just two actions forward:


What did you physically do to subtract \(\text3\) steps? You initially took \(\text5\) measures forward but then you take it \(\text3\) actions backward come land up earlier with only \(\text2\) measures forward. The backward displacement is represented by an arrow pointing to the left (backwards) with length \(\text3\). The net an outcome of adding these 2 vectors is \(\text2\) steps forward:


Thus, individually a vector from one more is the exact same as adding a vector in the contrary direction (i.e. Subtracting \(\text3\) steps forwards is the same as adding \(\text3\) steps backwards).