If you were to start right over here and you were to rotate around So if I were to start, if I were to, let me draw some coordinate axes here. So let's just first thinkĪbout what a negative 270 degree rotation actually is. So actually let me go over here so I can actually draw on it. The points of this triangle around the origin by negative 270 degrees, where is it gonna put these points? And to help us think about that, I have copied and pasted So what we want to do is think about, well look, if we rotate And this tool, I can put points in, or I could delete points. So positive is counter-clockwise, which is a standard convention, and this is negative, so a negative degree would be clockwise. The direction of rotationīy a positive angle is counter-clockwise. So this is the triangle PINĪnd we're gonna rotate it negative 270 degrees about the origin. We're told that triangle PIN is rotated negative 270ĭegrees about the origin. I hope this gives you more of an intuitive sense. If you want, you can connect each vertex and rotated vertex to the origin to see if the angle is indeed 90 degrees. As per the definition of rotation, the angles APA', BPB', and CPC', or the angle from a vertex to the point of rotation (where your finger is) to the transformed vertex, should be equal to 90 degrees. The rotated triangle will be called triangle A'B'C'. The point at which we do the rotation, we'll call point P. Well, let's say the shape is a triangle with vertices A, B, and C, and we want to rotate it 90 degrees. The shape is being rotated! But how do we do this for a specific angle? With your finger firmly on that point, rotate the paper on top. Now place your finger on the rotation point. Put another paper on top of it (I like to imagine this one as being something like a transparent sheet protector, and I draw on it using a dry-erase marker) and trace the point/shape. Here's something that helps me visualize it: The "formula" for a rotation depends on the direction of the rotation. I'm sorry about the confusion with my original message above. If you want to do a clockwise rotation follow these formulas: 90 = (b, -a) 180 = (-a, -b) 270 = (-b, a) 360 = (a, b). Also this is for a counterclockwise rotation. 360 degrees doesn't change since it is a full rotation or a full circle. 180 degrees and 360 degrees are also opposites of each other. So, (-b, a) is for 90 degrees and (b, -a) is for 270. The way that I remember it is that 90 degrees and 270 degrees are basically the opposite of each other.
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Read More Highly Skilled and Ready to Lead, Tuck’s Latest MBA Graduates Coveted by Top Firms*Length units are for your reference only since the value of the resulting lengths will always be the same no matter what the units are. The perimeter of an isosceles triangle is the sum of all its sides. For the third consecutive year-and ninth out of the last 10-95 percent or more of the latest Tuck graduates received a job offer within three months after graduation. Tuck graduates remain in high demand at top firms around the world. Once we know sides a, b, and c we can calculate the perimeter P, the semiperimeter s, the area K, and the altitudes: h a, h b, and h c. For example, if we know a and b we know c since c a. Highly Skilled and Ready to Lead, Tuck’s Latest MBA Graduates Coveted by Top Firms In our calculations for a right triangle we only consider 2 known sides to calculate the other 7 unknowns. However, it already has become apparent that this is not the whole story, because if you keep moving the speaker you again can achieve constructive interference. If the speakers are separated by half a wavelength, then there is destructive interference, regardless of how far or close you are to the speakers. If the speakers are at the same position, there will be constructive interference at all points directly in front of the speaker. What the example of the speakers shows is that it is the separation of the two speakers that determines whether there will be constructive or destructive interference. What happens if we keep moving the speaker back? At some point the peaks of the two waves will again line up:Īt this position, we will again have constructive interference! As the speaker is moved back the waves alternate between constructive and destructive interference. So, in the example with the speakers, we must move the speaker back by one half of a wavelength. If we look back at the first two figures in this section, we see that the waves are shifted by half of a wavelength. How far back must we move the speaker to go from constructive to destructive interference? We know that the distance between peaks in a wave is equal to the wavelength. In other words, the sound gets louder as you block one speaker! Equally as strange, if you now block one speaker, the destructive interference goes away and you hear the unblocked speaker. If we stand in front of the speakers right now, we will not hear anything! This must be experienced to really appreciate. Now imagine that we start moving on of the speakers back:Īt some point, the two waves will be out of phase – that is, the peaks of one line up with the valleys of the other creating the conditions for destructive interference. If we stand in front of the two speakers, we will hear a tone louder than the individual speakers would produce. If we place them side-by-side, point them in the same direction and play the same frequency, we have just the situation described above to produce constructive interference: The simplest way to create two sound waves is to use two speakers. We shall see that there are many ways to create a pair of waves to demonstrate interference. One wave alone behaves just as we have been discussing. The most important requirement for interference is to have at least two waves. We will explore how to hear this difference in detail in Lab 7. Given the fact that in one case we get a bigger (or louder) wave, and in the other case we get nothing, there should be a pretty big difference between the two. How could we observe this difference between constructive and destructive interference. Phase, itself, is an important aspect of waves, but we will not use this concept in this course. Similarly, when the peaks of one wave line up with the valleys of the other, the waves are said to be "out-of-phase". Although this phrase is not so important for this course, it is so commonly used that I might use it without thinking and you may hear it used in other settings. Often, this is describe by saying the waves are "in-phase". When the peaks of the waves line up, there is constructive interference. The sum of two waves can be less than either wave, alone, and can even be zero. In fact, at all points the two waves exactly cancel each other out and there is no wave left! This is the single most amazing aspect of waves. When the first wave is down and the second is up, they again add to zero. Now what happens if we add these waves together? When the first wave is up, the second wave is down and the two add to zero. However, carefully consider the next situation, again where two waves with the same frequency are traveling in the same direction: You may be thinking that this is pretty obvious and natural – of course the sum of two waves will be bigger than each wave on its own. The waves are adding together to form a bigger wave. This situation, where the resultant wave is bigger than either of the two original, is called constructive interference. If we add these two waves together, point-by-point, we end up with a new wave that looks pretty much like the original waves but its amplitude is larger. To start exploring the implications of the statement above, let’s consider two waves with the same frequency traveling in the same direction: As it turns out, when waves are at the same place at the same time, the amplitudes of the waves simply add together and this is really all we need to know! However, the consequences of this are profound and sometimes startling. Thus, we need to know how to handle this situation. This is very different from solid objects. In the last section we discussed the fact that waves can move through each other, which means that they can be in the same place at the same time. Constructive and Destructive Interferenceĥ.2 Constructive and Destructive Interference |
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