Engineering Mechanics | Adding and subtracting two vectors (Graphical Method)

Adding and subtracting two vectors (Graphical Method):

When we add two vectors clip_image001and clip_image002by graphical method to get clip_image003, we take vector clip_image001[1], put the tail of clip_image002[1]on the head of clip_image001[2].Then we draw a vector from the tail of clip_image001[3]to the head of clip_image002[2]. That vector represents the resultant clip_image003[1](Figure 4). I leave it as an exercise for you to show that clip_image004. In other words, show that vector addition is commutative.



Let us try to understand that it is indeed meaningful to add two vectors like this. Imagine the following situations. Suppose when we hit a ball, we can give it velocity clip_image002[3]. Now imagine a ball is moving with velocity clip_image006 and you hit it an additional velocity clip_image002[4]. From experience you know that the ball will now start moving in a direction different from that of clip_image007. This final direction is the direction of clip_image008and the magnitude of velocity now is going to be given by the length of clip_image008[1].

Now if we add a vector clip_image006[1] to itself, it is clear from the graphical method that its magnitude is going to be 2 times the magnitude of clip_image006[2] and the direction is going to remain the same as that of clip_image006[3]. This is equivalent to multiplying the vector clip_image006[4] by 2. Similarly if 3 vectors are added we get the resultant clip_image009. So we have now got the idea of multiplying a vector by a number n . If simply means: add the vector n times and this results in giving a vector in the same direction with a magnitude that n times larger.

You may now ask: can I multiply by a negative number? The answer is yes. Let us see what happens, for example, when I multiply a vector clip_image006[5] by -1. Recall from your school mathematics that multiplying by -1 changes the number to the other side of the number line. Thus the number -2 is two steps to the left of 0 whereas the number 2 is two steps to the right. It is exactly the same with vectors. If clip_image001[4]represents a vector to the right, clip_image010would represent a vector in the direction opposite i.e. to the left. It is now easy to understand what does the vector clip_image010[1]represent? It is a vector of the same magnitude as that of clip_image001[5]but in the direction opposite to it (Figure 5). Having defined clip_image010[2], it is now easy to see what is the vector clip_image011? It is a vector of magnitude clip_image012in the direction opposite to clip_image001[6].


Having defined clip_image010[3], it is now straightforward to subtract one vector from the other. To subtract a vector clip_image014 from clip_image006[6], we simply add clip_image015to clip_image006[7] that is clip_image016. Thus to subtract vector clip_image014[1]from clip_image006[8] graphically, we add clip_image006[9]and clip_image015[1]. This is shown in figure 6.


Again I leave it as an exercise for you to show that clip_image018is not equal to clip_image019butclip_image019[1]= – clip_image018[1]. We now solve a couple of examples.

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