Explanation Given: The line passes through the points ( 1 , 4 , 5 ) and ( − 2 , 3 , − 1 ) The second line is parallel to the first line and passes through the point ( 1 , 0 , 0 ) Formula used: The vector form of the equation of a line is given by, r ( t ) = r o + t v , Where r o is a point on the line and v is the vector parallel to the line The same can be used to find the line passing through two points, except here, the vector v that is parallel to the line is first deduced from the two points given. 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 → Vector operations Vector addition To add the vectors 〈 x 1 , y 1 〉 and 〈 x 2 , y 2 〉 , we add the corresponding components from each vector: 〈 x 1 + x 2 , y 1 + y 2 〉 . Scalar multiplication When we multiply a vector by a scalar, the result is a vector. When a vector is multiplied by a scalar i.e. a regular number, we have to multiply it on every component in the vector to get the resultant vector. Calculation: Let the given points P ( 1 , 4 , 5 ) , Q ( − 2 , 3 , − 1 ) and R ( 1 , 0 , 0 ) be written in the form of point vectors as O P → , O Q → and O R → respectively. Description P ( 1 , 4 , 5 ) , Q ( − 2 , 3 , − 1 ) and R ( 1 , 0 , 0 ) Step 1. To find the parametrization of the line passing through the two given points, the vector that is parallel to the line is first deduced from the two points given. Since the two points P ( 1 , 4 , 5 ) and Q ( − 2 , 3 , − 1 ) are on the line the vector between them will also lie on the line and will hence be parallel to the line. The vector P Q → can be found using the formula, P Q → = O Q → − O P → P Q → = O Q → − O P → = 〈 − 2 , 3 , − 1 〉 − 〈 1 , 4 , 5 〉 = 〈 − 2 − 2 , 3 − 4 , − 1 − 5 〉 = 〈 − 4 , − 1 , − 6 〉 Step 2...

4th Edition

ISBN: 9781319323394

Author: Rogawski

Publisher: MAC HIGHER

See similar textbooks

Concept explainers

Minimization

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in ter…

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which …

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the …

Videos

Question

Chapter 13, Problem 15CRE

To determine

To calculate:

A parametrization of the line passing through the given points

A parametrization of the line parallel to the given line and passing through a point

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 13, Problem 14CRE

Chapter 13, Problem 16CRE

Students have asked these similar questions

Use sigma notation to write the sum. Σ EM i=1 - n 2 4n + n n

Vectors t = 3i + 7j, u = 2i − 5j, and v = −21i + 9j are given.Part A: Find the angle between vectors t and u. Show all necessary calculations. Part B: Choose a value for c, such that c > 1. Find w = cv. Show all necessary work.Part C: Use the dot product to determine if t and w are parallel, orthogonal, or neither. Justify your answer.

A small company of science writers found that its rate of profit (in thousands of dollars) after t years of operation is given by P'(t) = (5t + 15) (t² + 6t+9) ³. (a) Find the total profit in the first three years. (b) Find the profit in the sixth year of operation. (c) What is happening to the annual profit over the long run? (a) The total profit in the first three years is $ (Round to the nearest dollar as needed.)

Chapter 13 Solutions

CALCULUS W/SAPLING ACCESS >IC<

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

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

A parametrization of the line passing through the given points A parametrization of the line parallel to the given line and passing through a point

Solution Summary: The author explains how to calculate a parametrization of the line passing through the points.

Concept explainers

Videos

To calculate: A parametrization of the line passing through the given points A parametrization of the line parallel to the given line and passing through a point

Want to see the full answer?

Chapter 13 Solutions

To calculate:

A parametrization of the line passing through the given points

A parametrization of the line parallel to the given line and passing through a point