Magnitudes of Vectors Don't Add UpDate: 11/14/2010 at 02:15:12 From: David Subject: Why does the vector law of addition work? I don't understand how adding vectors results in a triangle in which the third side is equivalent to the sum of the original two vectors. In particular, I don't understand how the sum of the two added vectors can have the same magnitude as the vector sum. A vector is defined as something with magnitude and direction, so vectors are equal if and only if they have the same magnitude and direction. The addition of vectors means combining two vectors, so the result of vector addition should give a vector with the same direction and magnitude as that of the combination of the added vectors, right? I can see how the sum of vectors A and B, if combined, would have the same direction as the third side of a triangle. What I don't understand here is how the magnitude of the third side can be equal to the magnitude of the other two sides. That would mean two sides of a triangle sum to the third side, wouldn't it? Date: 11/14/2010 at 10:01:22 From: Doctor Ian Subject: Re: Why does the vector law of addition work? Hi David, Suppose you are standing on a giant grid. You are given two numbers (a,b). You move a units to the east, and b units to the north. Now you are given two more numbers (c,d). You move c units to the east, and d units to the north. What is the total distance you've moved to the east? It's a + c, right? And what is the total distance you've moved to the north? It's b + d, right? So you could have got to the same final point by being given the numbers (a + c, b + d) along with the same instructions. Does this make sense? Do you see how it illustrates the rule for vector addition? > A vector is defined as something with magnitude and direction, so > vectors are equal if and only if they have the same magnitude and > direction. The addition of vectors means combining two vectors, so the > result of vector addition should give a vector with the same direction > and magnitude as that of the combination of the added vectors, right? Right. And they are combined by adding their components, as illustrated in the example above. > I can see how the sum of vectors A and B, if combined, would have the > same direction as the third side of a triangle. What I don't understand > here is how the magnitude of the third side can be equal to the > magnitude of the other two sides. That would mean two sides of a > triangle sum to the third side, wouldn't it? The magnitudes don't add directly. If you add two vectors, the magnitude of the resulting vector will be somewhere between zero and the sum of the individual magnitudes. The latter occurs when they have the same direction, e.g., (3,0) + (4,0) = (7,0) The vectors on the left have magnitudes of 3 and 4, and the sum has a magnitude of 7. The former occurs when they have opposite directions, but the same magnitude, e.g., (3,0) + (-3,0) = (0,0) The vectors on the left have magnitude 3, but they cancel each other out, leaving a null vector, with no direction or magnitude. In between, we might have something like (3,0) + (0,4) = (3,4) Here, the vectors on the left have magnitudes 3 and 4, but the sum has a magnitude of 5. That would correspond to a situation like Two guys are pushing on a box. One pushes to the east with a force of 3 lbs, while the other pushes to the north with a force of 4 lbs. What is the resultant force on the box? We can add the vectors to get (3,4). The magnitude of that is sqrt(3^2 + 4^2) = sqrt(25) = 5 The direction of that is tan^-1(4/3) = about 53 degrees So we could replace the two guys with one guy, pushing with a force of 5 lbs, at an angle of 53 degrees from the x-axis, and the box would move in the same way as when the two guys push it. Now, why doesn't the combined force have a magnitude of 7 lbs? Well, think of it this way: the box is moving at an angle to the force being applied by the guy pushing to the east. So only SOME of his force is going to moving the box. And the same is true for the guy pushing to the east. So we should expect the resultant force to be less than either of the individual forces. Now try thinking about those other kinds of cases. In one, the two guys are both pushing east, and their forces add up -- so the box moves to the east, under a force of 3 + 4 = 7 lbs. In the other, one guy is pushing east while the other pushes west, and since they apply the same magnitude of force, the box doesn't move at all. That is, it's like a force of 3 + -3 = 0 lbs is being applied to it. In terms of triangles, you can think of it this way. The hands of a clock always form two sides of a triangle, right? And the third side of that is the line connecting the hands. Would you expect the length of that third side to always be the sum of the lengths of the individual hands? Or does the angle between them have something to do with it? Does this help? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ Date: 11/15/2010 at 09:49:43 From: David Subject: Thank you (Why does the vector law of addition work?) Yes that clarified it, thank you!!! |
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