20.3 nature of vectors (ESAGN)

Vectors room mathematical objects and also we will now study several of their math properties.

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If 2 vectors have the very same magnitude (size) and the exact same direction, climate we speak to them same to each other. For example, if we have actually two forces, (vecF_1 = ext20 ext N) in the upward direction and (vecF_2 = ext20 ext N) in the upward direction, climate we have the right to say the (vecF_1 = vecF_2).

Equality that vectors

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

Just like scalars which have the right to have hopeful or an adverse values, vectors can also be optimistic or negative. A an unfavorable vector is a vector i m sorry points in the direction opposite to the reference hopeful direction. Because that example, if in a details situation, we specify the increase direction as the reference hopeful direction, then a force (vecF_1 = ext30 ext N) downwards would be a negative vector and could additionally be created as (vecF_1 = - ext30 ext N). In this case, the negative sign ((-)) indicates that the direction that (vecF_1) is opposite to the of the reference positive direction.

an adverse vector

A an unfavorable vector is a vector that has actually the opposite direction come the reference hopeful direction.

Like scalars, vectors can also be included and subtracted. We will certainly investigate exactly how to carry out this next.

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Addition and subtraction that vectors (ESAGO)

including vectors

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

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


It is an extremely easy to know the ide 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 that points native the initial place to the last position.

Adding vectors


masking tape


Tape a heat of masking tape horizontally throughout the floor. This will be your beginning point.

Task 1:

Take ( ext2) actions in the front direction. Use a item of masking ice to mark your end suggest and label it A. Climate take an additional ( ext3) actions in the forward direction. Usage masking ice to note your last position together B. Make certain you shot to save your actions all the same length!

Task 2:

Go ago to your starting line. Now take ( ext3) actions forward. Use a piece of masking ice to mark your end point and label it B. Climate take an additional ( ext2) procedures forward and use a brand-new piece that masking ice cream to mark your last position together A.


What do you notice?

In Task 1, the very first ( ext2) steps forward represent a displacement vector and the 2nd ( ext3) actions forward also form a displacement vector. If we did not prevent after the very first ( ext2) steps, us would have actually taken ( ext5) procedures in the front direction in total. Therefore, if we include the displacement vectors for ( ext2) steps and ( ext3) steps, we should acquire a total of ( ext5) actions in the forward direction.

It does not matter whether you take it ( ext3) measures forward and also then ( ext2) steps forward, or 2 steps complied with by an additional ( ext3) steps forward. Your last position is the same! The order of the enhancement does not matter!

We can represent vector addition graphically, based upon the task above. Attract the vector because that the very first two actions forward, complied with by the vector through the following three actions forward.


We include the second vector at the finish of the an initial vector, due to the fact that this is whereby we now are ~ the an initial vector has actually acted. The vector native the tail that the first vector (the starting point) to the head the the 2nd vector (the finish point) is then the amount of the vectors.

As you have the right to convince yourself, the order in i beg your pardon you add vectors does not matter. In the example above, if you made decision to very first go ( ext3) measures forward and then an additional ( ext2) measures forward, the end result would still it is in ( ext5) steps forward.

individually vectors

Let"s go back to the difficulty of the hefty box that you and your friend room trying come move. If friend didn"t interact properly first, friend both could think that you should 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 girlfriend stands in ~ the former of the box and also pulls it towards them with a pressure (vecF_2). In this instance the two pressures are in opposite directions. If we specify the direction your friend is pulling in as positive then the force you are exerting should be negative due to the fact that it is in opposing direction. We can write the total force exerted on package as the sum of the separation, personal, instance forces:


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

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As we did before, we have the right to illustrate vector individually nicely using displacement vectors. If you take ( ext5) steps forward and also then subtract ( ext3) procedures forward you are left with just two actions forward:


What did you physically execute to subtract ( ext3) steps? You originally took ( ext5) measures forward but then you took ( ext3) actions backward come land up back with just ( ext2) procedures forward. That backward displacement is represented by an arrow pointing to the left (backwards) with size ( ext3). The net an outcome of adding these 2 vectors is ( ext2) procedures forward:


Thus, subtracting a vector from one more is the exact same as adding a vector in the contrary direction (i.e. Subtracting ( ext3) procedures forwards is the same as including ( ext3) measures backwards).