Explanation Given: The following points, A = ( 2 , − 1 ) B = ( 1 , 4 ) P = ( 2 , 3 ) Vector A B → is equivalent to the vector P Q → , Key concepts applied: Vector joining two points If P 1 ( x 1 , y 1 ) and P 2 ( x 2 , y 2 ) are any two points, then the vector joining P 1 and P 2 is given by, P 1 P 2 → = O P 2 → − O P 1 → Components of a vector In a two-dimensional coordinate system, any vector can be broken into x- component and y -component. For example, in the vector, v → = 〈 v x , v y 〉 , v x is the x- component and v y is the y -component. Consider a vector r written as x i + y j . This form of the vector is called its component form. Here, x and y are called as the scalar components of the vector, and x i , and y j and are called the vector components of the vector along the respective axes. Sometimes x and y are also termed as the rectangular components. Length of a vector The magnitude of a vector is the length of the vector. The length of a vector in Cartesian coordinates is the square root of the sum of the squares of its coordinates. For a two-dimensional vector, the length of the vector is found using the following formula. ‖ a ‖ = a x 2 + a y 2 Equivalent vectors Two arrows that are pointing in the same direction, and the same length, but are located at different positions, represent vectors that are indistinguishable from one another, because they have the same magnitude and direction. We say that these vectors are equivalent, or equal. A vector in the plane is a directed line segment. Two vectors are equivalent if they have the same magnitude and direction. Note that the equivalence of vectors requires only the same magnitude and the same direction—not the same location. Conditions of vectors equality Vectors are equal if their coordinates or components are equal. Formula used: Vector joining two points If P 1 ( x 1 , y 1 ) and P 2 ( x 2 , y 2 ) are any two points, then the vector joining P 1 and P 2 is given by, P 1 P 2 → = O P 2 → − O P 1 → Components of a vector In a two-dimensional coordinate system, any vector can be broken into x- component and y -component. For example, in the vector, v → = 〈 v x , v y 〉 , v x is the x- component and v y is the y -component. Length of a vector For a two-dimensional vector, the length of the vector is found using the following formula. ‖ a ‖ = a x 2 + a y 2 Direction of equivalent vectors If the lines that they are on all have the same slope, the vectors have the same direction. If P 1 ( x 1 , y 1 ) and P 2 ( x 2 , y 2 ) are any two points, the slope of the line joining them P 1 P 2 → are given by, S l o p e = tan θ = y 2 − y 1 x 2 − x 1 , where θ is the angle between the line and the x- axis Calculation: Let the points A , B , P and Q be written as the point vectors O A → , O B → , O P → and O Q → respectively...

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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System that uses coordinates to uniquely determine the position of points. The most common coordinate system is the Cartesian system, where points are given by distance along a horizontal x-axis and vertical y-axis from the origin. A polar coordinate system locates a point by its direction relative to a reference direction and its distance from a given point. In three dimensions, it leads to cylindrical and spherical coordinates.

Chapter 13, Problem 8CRE

To determine

The point Q

The sketch of the given vectors

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Chapter 13 Solutions

Calculus - Standalone book

Ch. 13.1 - Prob. 1PQ Ch. 13.1 - Prob. 2PQ Ch. 13.1 - Prob. 3PQ Ch. 13.1 - Prob. 4PQ Ch. 13.1 - Prob. 5PQ Ch. 13.1 - Prob. 6PQ Ch. 13.1 - Prob. 1E Ch. 13.1 - Prob. 2E Ch. 13.1 - Prob. 3E Ch. 13.1 - Prob. 4E

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The point Q The sketch of the given vectors

Solution Summary: The author explains how a vector can be divided into x- and y-components.