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  • These aren't locations anymore but ''vectors''. This number is then computed from these two ''vectors'':
    113 KB (19,680 words) - 00:08, 23 February 2019
  • ==The length of vectors, the norm== [[Image:length and angle.jpg|150px|right]]
    21 KB (3,396 words) - 20:31, 10 August 2011
  • We transition to from numbers to ''vectors''. But what are the operations? ...mer. This is very similar to multiplication of numbers; after all they are vectors of dimension $1$... Let's match the two problems and their solutions:
    113 KB (18,750 words) - 02:33, 10 December 2018
  • ...an listen to a conversation at the other focus even if there are obstacles between them. $\square$ ...ppens when the ''angular velocity'' is constant: the object turns the same angle per unit of time:
    130 KB (22,842 words) - 13:52, 24 November 2018
  • '''Exercise.''' What is the relation, if any, between these two pairs of properties: (a) even and one-to-one, (b) odd and one-to- Now, we know that the axis of a parabola lies half-way between the two $x$-intercepts. Therefore, it is:
    143 KB (24,052 words) - 13:11, 23 February 2019
  • Now, if the vectors represent velocities of particles, what kind of flow is this? This isn't a Now, if the vectors represent velocities of particles, what kind of flow is this? It looks like
    74 KB (13,039 words) - 14:05, 24 November 2018
  • The equations mean that the vectors of the vector field are tangent to these trajectories. ODEs are discussed i ...apter 21 that ''Newton's Law of Gravity'' states that the force of gravity between two objects is given by the formula:
    91 KB (16,253 words) - 04:52, 9 January 2019
  • *the difference between the two: $x_n-p_n$ and $y_n-q_n$, which is the direction from the hound tow *the unit vector in this direction (dividing by the distance between them);
    50 KB (8,692 words) - 14:29, 24 November 2018
  • Of course, it is sufficient to know the sign of the derivative to distinguish between increasing and decreasing behavior. Therefore, this behavior depends only o ...eveloping, the sign of the exterior derivative will tell us the difference between increasing and decreasing behavior. But the derivative only uses the topolo
    42 KB (7,131 words) - 17:31, 30 November 2015
  • ...''' only, until now. By that I mean that we did a lot of computations with vectors. The distance between $a=(a_1,\ldots,a_n)$ and $b=(b_1,\ldots,b_n)$ is
    14 KB (2,404 words) - 15:04, 13 October 2011
  • *the lengths of vectors and *the angles between vectors. $\\$
    35 KB (5,871 words) - 22:43, 7 April 2016
  • When a ball is thrown in the air under an angle, it moves in both vertical and horizontal directions, simultaneously and in The ''path'' of the ball will appear to an observer -- from the right angle -- as a curve. It is placed in the $xy$-plane positioned vertically:
    76 KB (13,017 words) - 20:26, 23 February 2019
  • ...is is very similar to multiplication of numbers; after all they are column-vectors in dimension $1$... Let's align the two problems: In other words, we need to find a way to stretch these two vectors so that the resulting combination is the vector on the right:
    46 KB (7,625 words) - 13:08, 26 February 2018
  • ...ling'': the rate of cooling of an object is proportional to the difference between its temperature and the ambient temperature. To sum up, the amount of heat exchanged between two rooms is proportional to:
    39 KB (6,850 words) - 15:29, 17 July 2015
  • *Find the plane through the point $P=(-1,6,-5)$ and parallel to the vectors $A=<1,1,0>$ and $B=<0,1,1>.$ *Vectors $A$ and $B$ are given below. Copy the picture and illustrate graphically (a
    46 KB (8,035 words) - 13:50, 15 March 2018
  • <!--s-->[[image:algebra of vectors.png|center]] *For every two vectors, what is their sum?
    13 KB (2,233 words) - 14:41, 20 February 2015
  • ...of the vector $N$ is revealed once we remember that the dot product of two vectors is $0$ if and only if they are perpendicular. [[image:plane as perpendicular vectors.png| center]]
    97 KB (17,654 words) - 13:59, 24 November 2018
  • Setting aside possible connections between the integrands, the pattern of the ''domains of integration'' is clear. The ...ion between $R$ and $\partial R$ is a matter of ''topology''. The relation between $d \omega$ and $\omega$ is a matter of ''calculus'', the calculus of differ
    34 KB (5,619 words) - 16:00, 30 November 2015
  • The distance between $a=(a_1,\ldots,a_n)$ and $b=(b_1,\ldots,b_n)$ is But does the angle between two vectors even make sense in ${\bf R}^n$?
    32 KB (5,426 words) - 21:57, 5 August 2016
  • The distance between $a=(a_1,\ldots,a_n)$ and $b=(b_1,\ldots,b_n)$ is We can measure vectors and, therefore, can consider [[convergence]]:
    2 KB (410 words) - 15:09, 9 June 2012
  • Of course, it is sufficient to know the sign of the derivative to distinguish between increasing and decreasing behavior. Therefore, this behavior depends only o ...eveloping, the sign of the exterior derivative will tell us the difference between increasing and decreasing behavior. But the derivative only uses the topolo
    41 KB (6,928 words) - 17:31, 26 October 2015
  • *Find the exact values of the $x$-coordinates of the intersections between the parabola and the line below: [[image:area between parabola and horizontal line.png| center]]
    17 KB (2,933 words) - 19:37, 30 July 2018
  • ...d by how much. To compare, we can measure lengths of vectors and the angle between them. [[Image:length and angle.jpg|250px]]
    3 KB (391 words) - 14:22, 26 July 2012
  • ...the velocity field (it is explained in this section). The main difference between the first and the rest is that the parametric curve isn't the ''integrand'' where $\alpha$ is the angle between $V$ and $D$.
    12 KB (2,194 words) - 14:37, 5 December 2017
  • The angle between $u$ and $v$ is $\alpha$ as: ${\rm cos \hspace{3pt}} \alpha = \frac{< u , v> Best in what sense? We mean that the "area" between the two graphs, $\sin$ and $g$ is minimized. In other words, minimize
    10 KB (1,688 words) - 17:59, 13 October 2011
  • ...tyle\lim_{x \rightarrow a} f(x) = L$. To illustrate, you don't have to use vectors... ...last condition encodes the idea of approximation: not only the difference between $f$ and its approximation is "small" but it's small even relative to the ch
    34 KB (5,665 words) - 15:12, 13 November 2012
  • '''MTH 231 - Calculus with Analytic Geometry III.''' Vectors, curves, and surfaces in space. Derivatives and integrals of functions of m ...he surface on the building. Also, the intersection point of all the normal vectors will be the focus of the mirror/window.
    10 KB (1,596 words) - 13:34, 27 November 2017
  • ...screte calculus]] we need to be able compute lengths of vectors and angles between them. An inner product is how one adds geometry to a [[vector space]]. ...er product'' on $V$ is a function that associates a number to each pair of vectors in $V$:
    4 KB (749 words) - 20:12, 1 May 2013
  • The amount of heat exchanged between two rooms is proportional to: #and, inversely, to the distance between the [[center of mass|centers of mass]] of the rooms.
    2 KB (311 words) - 13:17, 28 August 2015
  • '''MTH 231 - Calculus with Analytic Geometry III.''' Vectors, curves, and surfaces in space. Derivatives and integrals of functions of m 12.1 Vectors in the Plane,
    5 KB (621 words) - 14:57, 5 May 2014
  • '''MTH 231 - Calculus with Analytic Geometry III.''' Vectors, curves, and surfaces in space. Derivatives and integrals of functions of m 12.1 Vectors in the Plane,
    6 KB (805 words) - 13:38, 6 May 2015
  • *[[angle between vectors|angle between vectors]]
    16 KB (1,773 words) - 00:41, 17 February 2016
  • 6.1 Area Between Two Curves 12.1 [[Vectors]] in the Plane
    6 KB (634 words) - 16:38, 1 March 2013
  • (Note that the idea of decomposition of vectors is routinely used in physics, e.g., in the study of motion of an object on ...It is done with the help of the rotation matrix (shown at the top) for an angle of rotation $\alpha$:
    14 KB (2,504 words) - 14:59, 17 September 2019
  • ...heir “counterparts” in $K$ are also homologous. What is the correspondence between the finer cycles in $K'$ and the coarser cycles in $K$? ...using a polar coordinate parametrization, then measuring the change in the angle, and setting
    41 KB (7,169 words) - 14:00, 1 December 2015
  • ...(b) State the Cauchy-Schwarz inequality. (c) Define the angle between two vectors in an inner product space. Prove that it's well-defined. ...an element of an inner product space $V$ and suppose $S$ is the set of all vectors orthogonal to $a$, plus $0$. Prove that $S$ is a subspace of $V$.
    2 KB (376 words) - 20:27, 13 June 2011
  • $\bullet$ '''3.''' Vectors $A$ and $B$ are given below. Copy the picture and illustrate graphically: ( $\bullet$ '''4.''' Find the angle between the vectors $<1,1,1>$ and $<1,0,0>$. Don't simplify.
    2 KB (308 words) - 23:06, 14 March 2018
  • ...know the sign of the derivative or the exterior derivative to distinguish between increasing and decreasing behavior. But the latter only uses the topologica ...ch location in some region $D$ (the domain) in this space and each pair of vectors (directions) at that location:
    9 KB (1,604 words) - 18:08, 27 August 2015
  • 1. Vectors $a$ and $b$ are given below. Copy the picture and illustrate graphically (a 2. Find the angle between the vectors $<1,1,1>$ and $\mathbf{i}$. Don't simplify.
    799 bytes (132 words) - 15:21, 9 March 2014
  • The question is: what is the relation between the topology of the $xy$-plane and the topologies of the $x$- and $y$-axes? where $\phi$ is the angle of the rod and $R$ is the (fixed) length of the rod.
    44 KB (7,951 words) - 02:21, 30 November 2015
  • ...R}^n$: $E_1=<1,0,0,...,0>$ and $E_2=<0,1,0,...,0>$. (a) What is the angle between them? (b) What is the magnitude of the vector $E_1-E_2$? ...atisfy: $AB=I_2$, where $I_2$ is the identity matrix. What is the relation between $A$ and $B$? Explain.
    2 KB (351 words) - 20:49, 30 April 2018
  • ...'' Given vectors $a=<1,2>,\ b=<-2,1>$, find their magnitudes and the angle between them.
    2 KB (389 words) - 19:13, 6 May 2016
  • (Note that $f'(a), g'(a)$ are vectors, $f(a), g(a)$ are scalars) ...rphi {\rm \hspace{3pt} is \hspace{3pt} the \hspace{3pt} angle \hspace{3pt} between \hspace{3pt}} e {\rm \hspace{3pt} and \hspace{3pt}} f'(a), \\
    6 KB (962 words) - 15:45, 17 August 2011