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Determine whether the given vectors are orthogonal, parallel, or neither.
(a) $ u = \langle -5, 4, -2 \rangle $ , $ v = \langle 3, 4, -1 \rangle $(b) $ u = 9i - 6j + 3k $ , $ v = -6i + 4j - 2k $(c) $ u = \langle c, c, c \rangle $ , $ v = \langle c, 0, -c \rangle $
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Vectors and the Geometry of Space
The Dot Product
Oregon State University
University of Nottingham
Idaho State University
In mathematics, a vector (from the Latin word "vehere" meaning "to carry") is a geometric entity that has magnitude (or length) and direction. Vectors can be added to other vectors according to vector algebra. Vectors play an important role in physics, engineering, and mathematics.
In mathematics, a vector (from the Latin word "vehere" which means "to carry") is a geometric object that has a magnitude (or length) and direction. A vector can be thought of as an arrow in Euclidean space, drawn from the origin of the space to a point, and denoted by a letter. The magnitude of the vector is the distance from the origin to the point, and the direction is the angle between the direction of the vector and the axis, measured counterclockwise.
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this problem. We have three pairs of vectors we're going to examine. I've done one of them in black one and red and one in green. And for each pair of vectors we want to compare them. And we want to know, Are they orthogonal? Which means perpendicular to each other. Are they parallel vectors or neither? So how can we tell what orientation these vectors are to each other? We have a the're, um that's the cosine of Fada fate of being the angle between two vectors. The cosine of Fada is the dot product of the vectors divided by the product of the magnitudes of the vectors. We can use this. The're, um, to answer the question in this problem. First of all, let's look at orthogonal. If the vectors are parallel, that means that they're going to make a 90 degree angle so they they will be 90 degrees well, co sign of 90 degrees zero. So the on leeway for this fraction in this the're, um, toe equal to zero is if our numerator is equal to zero, which means that our dot product would have to equal zero. So let's start there. I'm gonna look in each case and see if they're worth again. Let's just do that first. So I'm going to find the dot product of each of these pairs. First, let's start with the black one. The dot product you dot v. Well, remember for a dot product, we're going to multiply the exes, multiply the wise multiply disease, and we're gonna add them up. So for the first one, when I multiply the exes, I get negative 15 multiplying. The wise gives me 16 multiplying disease gives me too. And that gives me a dot product of three. So these are not orthogonal. Okay, So I I know that to start with, not orthogonal for the black. What about my red pair? Well, this dot product when I multiply, uh, this case because I'm giving it of unit in eyes. Jason case, we still do the same thing on a multiply the coefficients for the eyes, jays and case separately and add them up. I get negative 54 minus 24 minus six. And that gives me a dot product of negative 84. This also is not orthogonal. What about the green ones? Well, this dot product when I multiply the exes. That gives me C squared. The wise gives me zero and disease. Give me negative C squared. That is zero. So my green pear, This is an orthogonal pair avert for of an orthogonal pair of vectors. Okay, Now let's take a look at our second case. Parallel. Well, if two vectors are parallel to each other, then if I lined them up next to each other, the angle between them is either going to be. Zero degrees means they're on top of each other, pointing the same direction or 180. They line up there pointing in opposite directions. So if my ankle is zero or 1 80 then CoSine is either going to be one or negative one. Okay, so which means that that dot product, that numerator and denominator have to either be identical or they have to be opposite sides if either of that is true. So if I get two and two, that's gonna be parallel. If I get to a negative two, that's parallel as well, so I already have the dot products put together. So let's make our denominator. Let's find the magnitude of each of the vectors and we'll add them together. So we'll start with our black pair. The magnitude of vector you Well, that's going to be the square root of each of these numbers. Um, squared. So that's going to be 25 plus 16 plus four, which gives me a square root of the square root of 45. How about V? Well, that magnitude is going to be the square root of nine plus 16 plus one, which is going to give me the square root of 26. Now, if I add if I multiply those together square to 45 times the square to 26 I don't even have to do this to see that it is not going to equal three or negative three. So that numerator denominator. You know, when I compare that to three there, this is not gonna work. This is neither. It is not orthogonal. It is not parallel. Hey, don't have to test our greens. We already know that those air perpendicular to each other. So what about the red pair? Well, the magnitude of you that's going to be the square root of 81 plus 36 plus nine and that's going to give me the square root of 126 square root of V that's going to be 36 plus 16 plus four under that radical, and that's going to give me a value of the square root of 56. Now this one, we do need to multiply together. I can't do this one in my head. And when I multiply those two magnitudes together, that gives me the square root of 7056 which equals 84 so that give me a denominator of 84 a numerator of negative 84 value of negative one. That is a parallel set of vectors. So those are my three pairings and our results.
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