Explanation Given: The point P = ( 1 , 4 , − 3 ) The vector P Q → = 〈 3 , − 1 , 5 〉 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 three-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 z 〉 , v x is the x- component, v y is the y -component and v z is the z -component. 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 Unit vector A vector whose magnitude is unity (i.e., 1 unit) is called a unit vector. The unit vector in the direction of a given vector a (often denoted with a cap a ^ ) is obtained by dividing the vector with its length. a ^ = a ‖ a ‖ Formula used: For a three-dimensional vector, the length of the vector is found using the following formula. ‖ a ‖ = a x 2 + a y 2 + a z 2 The unit vector in the direction of a given vector a (often denoted with a cap a ^ ) is obtained by dividing the vector with its length. a ^ = a ‖ a ‖ Calculations: Let the points P and Q be written as the point vectors O P → and O Q → respectively. Let Q be the point ( x , y ) . Description P = ( 1 , 4 , − 3 ) , P Q → = 〈 3 , − 1 , 5 〉 Step 1...

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ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Vector calculus is an important branch of mathematics and it relates two important branches of mathematics namely vector and calculus.

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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 12CRE

To determine

To calculate:

The point that satisfies the given condition

The unit vector equivalent to the given vector

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Chapter 13, Problem 11CRE

Chapter 13, Problem 13CRE

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Chapter 13 Solutions

CALCULUS (CLOTH)

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 that satisfies the given condition The unit vector equivalent to the given vector

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

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To calculate: The point that satisfies the given condition The unit vector equivalent to the given vector

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Chapter 13 Solutions

To calculate:

The point that satisfies the given condition

The unit vector equivalent to the given vector