Dot-product-of-two-parallel-vectors

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Lines, Parallel and Perpendicular Lines, Parallel and Perpendicular

In mathematics, the term "straight line" is one of the few terms that is left undefined. However, most people are comfortable with this undefined concept, which can be modeled by a pencil, a stiff wire, the edge of a ruler, or even an uncooked piece of spaghetti. Mathematicians sometimes think of a line as a point moving forever through space. Lines can be curved or straight, but in this entry, only straight lines are considered. A line, in the language of mathematics, has only one dimension—length—and has no end. It stretches on forever in both directions, so that its length cannot be measured. When a line is modeled with a piece of spaghetti, a line segment is actually being represented. The model of a line segment has thickness (or width), while the idea that it models—a mathematical line—does not. So a mathematical line is a notion in one's mind, rather than a real object one can touch and feel, just as the notion of "two" is an idea in one's mind—that quality and meaning that is shared by two apples, two trucks, and the symbols //, 2, ☺☺, and ii. Think of two straight lines in a plane (another undefined term in geometry ). Someone can model this idea, imperfectly, by two pencils or two pieces of spaghetti lying on a desktop. Now, mentally or on a desktop, push these lines around, still keeping them on the plane, and see the different ways two lines can be arranged. If these two lines meet or cross, they have one point in common. In the language of mathematics, the two lines intersect at one point, their point of intersection. If two lines are moved so that they coincide, or become one line, then they have all of their points in common. What other arrangements are possible for two lines in a plane? One can place them so that they do not coincide (that is, one can see that they are two separate lines), and yet they do not cross, and will never cross, no matter how far they are extended. Two lines in the same plane, which have no point in common and will never meet, are called parallel lines. If one draws a grid, or coordinate system, on the plane, she can see that two parallel lines have the same slope, or steepness. Are there any parallel lines in nature, or in the human-made world? There are many models of parallel lines in the world we build: railroad tracks, the opposite sides of a picture frame, the lines at the corners of a room, fence posts. In nature, parallel lines are not quite so common, and the models are only approximate: tracks of an animal in the snow, tree trunks in a forest, rays of sunlight. The only other possible arrangement for two lines in the plane is also modeled by a picture frame, or a piece of poster board. Two sides of a rectangle that are not parallel are perpendicular . Perpendicular lines meet, or intersect, at right angles, that is, the four angles formed are all equal. The first pair of lines in part (a) of the figure below meet to form four equal angles; they are perpendicular. The second pair in part (b) forms two larger angles and two smaller ones; they are not perpendicular. Perpendicular lines occur everywhere in buildings and in other constructions. Like parallel lines, they are less common in nature. On a coordinate system, two perpendicular lines (unless one of them is horizontal) have slopes that multiply to a product of -1; for example, if a line has a slope of 3, any line perpendicular to it will have a slope of -⅓. see also Lines, Skew; Slope. Lucia McKay Anderson, Raymond W. Romping Through Mathematics. New York: Alfred A. Knopf, 1961. Juster, Norton. The Dot and the Line: A Romance in Lower Mathematics. New York: Random House, 1963. Konkle, Gail S. Shapes and Perception: An Intuitive Approach to Geometry. Boston: Prindle, Weber and Schmidt, Inc., 1974.


From Omilili

Dot Product Definition of a Plane?

In? I understand the plane in itself can be defined through the dot product (whenever a vector is perpendicular... of the normal on the other vector) would be zero... Unless the d expresses the placement of the plane ?\langle N, (X - P)\rangle = 0 . You want that if I draw a line from any place in the plane (X) to a point

From Yahoo Answers

Question:How could I find the pitch between two points in 3D space. (The angle between two vectors when you look at them from the side.) I need the formula for a physics engine to be used in a game so I would like to know how i could implement the formula in c-style code. Thanks, Tom.

Answers:Use the dot product formula: P.Q=|P||Q|cos =arccos((vector P . vector Q)/(magnitude P * mangitude Q)) where vectors are magnitude times direction in i,j,k components

Question:Here are two vectors: a = (3.0 m)i - (3.0 m)j and b = (3.0 m)i + (5.0 m)j. What is the magnitude of a? What is the angle of a(relative to i)? What is the magnitude of b - a? What is the angle of b - a? What is the magnitude of a - b? What is the angle of a - b? I'm somewhat confused on how to work these parts of the problem. Thanks for any insight or help you can offer. Maybe I should elaborate some... i is the x component of a graph j represents the y component we are supposed to use the pythagorean theorem to find the magnitude and trigonometric functions to find the angles...kinda stumped though

Answers:magnitude is just the square root of the componets squared: ||A|| = sqrt(3 + 3 ) = sqrt(18) For the angle use the dot product: c d = ||c|| ||d|| cos for a relative to i <3,-3><1,0> = sqrt(18)*1 cos 3 = sqrt(18) cos = 45 b - a: <3,5> - <3,-3> = <0,8>

Question:

Answers:Do they intersect? And what are their relative directions?


From Youtube

Lecture 03_ Vectors - Dot Products - Cross Products - 3D Kinematics.mp4



Geometric and algebraic definition of dot product

Explain the algebraic and geometrical connection of vector dot product. Reference to Howard Anton's Calculus text

Scalar Product and Vector Product Of two Vectors

Check us out at www.tutorvista.com Scalar Product : The scalar product and the vector product are the two ways of multiplying vectors which see the most application in physics and astronomy. The scalar product of two vectors can be constructed by taking the component of one vector in the direction of the other and multiplying it times the magnitude of the other vector

Calculating dot and cross products with unit vector notation

Calculating the dot and cross products when vectors are presented in their x, y, and z (or i,j, and k) components.


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